Great question. I have divided my answer into sections: the first three address the three aspects of the question and answers them with a direct approach, the fourth analyzes the nature of the question itself and how this suggests a deeper explanation for the answer, and the fifth is a conclusion to state my answer more clearly and succinctly. If my answer is unclear at any point, please feel free to comment asking for clarification and/or suggesting an edit.
A formal definition of intelligence is something that's been attempted by and debated between philosophers for as long as the concept has existed. There is not a general consensus, and this is for several reasons, more than could possibly be listed here. I would argue however that the two biggest reasons are 1) the fact that how we understand intelligence (much less define it) is very subjective and arbitrary, and 2) the inability to distinct between "true" intelligence and the appearance of intelligence. I think these two reasons are sufficient in answering your question and I will talk about both of them below:
The assignment of intelligence is subjective and arbitrary. For example, dirt is generally considered by humans to not be intelligent. Now say Earth is discovered by a living thing that is not carbon based and has evolved in such a way that it is physically unrecognizable to us as life, but satisfies some hypothetical abstract definition such that it is intelligent. This creature would equally not recognize us as living things. Similarly, let's say that it's conception of society, language, et cetera, is also fundamentally different to ours and mutually unrecognizable. This creature would simply see us as objects which move around entirely as a result of physical laws, void of motivation or intent. We may as well be dirt to this creature. Therefore, while we each recognize ourselves as intelligent, we don't recognize the other as such. So we see that what we assign as intelligent or not is entirely dependent of our subjective experience, and therefore a general definition of intelligence, independent of the human experience (which would expose a specific nature or identity, as you said in your question), is incredibly difficult to formulate, if not impossible. Furthermore, the assignment of intelligence is also incredibly arbitrary (making it meaningless), in that there must be a cut off of sorts. That is, we recognize certain things as more or less intelligent as compared to each other, but we also recognize things as having no intelligence. This implies that there exists a cut off, as in a minimum wherein things above it are intelligent and things below aren't. If we see intelligence as a property which exists on a spectrum, the cut off point is clearly arbitrary. If we see it as a set of qualities, which may or may not exist on a spectrum, and we say that if something is absent of one of these qualities then it is not intelligent, then it's clear to see that the qualities themselves are either arbitrary or solely dependent on the subjective human experience, as aforementioned. Therefore, it's impossible to construct a general, abstract definition of intelligence which is void of a bias towards the human experience, but also encompasses what humans generally do or don't consider intelligent.
It is impossible to distinct between "true"(natural) intelligence and the appearance of intelligence (artificial intelligence). I think this issue is explained well by the following thought experiment: Say you're in a prison cell. You stay here indefinitely with no form of communication to the outside world and there is a prisoner in an identical cell next to you under those same conditions. There is a miniscule gap in the wall adjoining your cells and you each have paper and pencil, by which notes may be passed through the gap. You can not speak aloud, however. The other prisoner writes in a language you can't read, say Mandarin. But, there is a book in your cell which contains Mandarin phrases and what phrases to respond with, in Mandarin. Using this book, you can effectively have a conversation in Mandarin without knowing the language, nor the content of the conversation. If you're the prisoner which speaks Mandarin, how could you ever claim that the other prisoner doesn't? How could you say they aren't intelligently deciding how to reply to your messages? This is, in essence, a metaphor for AI, such as the ChatGPT example you gave. The AI, when given an input, will give an output strictly according to its programming. Could you say that the AI "thought" about what to output? Could you even say that it "decided", much less "intelligently" decided, what to output? This, however, does not seem any different from how humans "decide" what to do(output) in a certain situation(input). Is the human brain not, in essence, functionally equivalent to the AI's programming, or furthermore, to the prisoner's book? The problem is: how could we, in general, distinct when an unknown intelligent agent has natural or artificial intelligence? It seems that we can't.
Your question doesn't require such a distinction, and even lists examples from both natural and artificial intelligence, but the absence of such a distinction has important implications which are very relevant to your question. I outline this below and continue to label it as point 2.
- In this case, the intelligence of these agents seems to be able to be simply broken down into the ability to output something given an input and a set of rules. This would however imply that all matter is inherently intelligent. For example, a rock in space, when input with a collision from another rock, will output a new velocity and possibly a "scar" of some sort. So then it seems that the intelligence of an agent should be defined by the degree of complexity of the rule set it uses to decide outputs. You may argue that we consider some AIs to be more intelligent than others, not because of the complexity of their programming but by how sensical their outputs are. However, this requires a standard for what "makes sense", which leads to the problem of subjectivity as outlined in the previous point. So then it seems the degree of complexity should be the deciding factor. But this leads to the problem of arbitrarity, which was also outlined in the previous point. Furthermore, isn’t the rule set of the rock the same as that of the human brain? That being the laws of physics? So then, not only is everything intelligent but everything's equally intelligent? This would make the the assignment of intelligence pretty meaningless. To resolve this, the concept of nonphysical rules can be introduced. That is, things outside of physical laws which an agent may possess that impacts how it makes decisions (e.g. free will). Personally, I would argue that these rules can't exist since something nonphysical impacting the physical world would violate conservation laws. Regardless, it can't be known whether an agent possesses a nonphysical rule or not since it inherently can't be measured. Therefore, a definition of intelligence which requires the existence or absence of nonphysical rules within an agent can't be known to encompass anything, much less the things humans typically consider intelligent.
In short, it is impossible to construct a meaningful definition of intelligence such that it encompasses the agents which humans typically call intelligent, but is abstract enough such that it is independent of the exact nature of the agent in question. This is by virtue of the definition being necessarily human centric, and therefore it's application to an agent depends on that agent's "similarity" to humans, i.e. the assignment of intelligence would depend on the nature of the agent in question. I would argue from here that intelligence is an entirely relative thing; an agent is intelligent at something or in terms of something, but not simply "intelligent". Elaborating on and more properly defining what that means, however, is not relevant to answering this question and would deserve its own discussion.
Claiming something is designed implies two things: That there is motivation/intent behind its design, and that its design has utility value to its designer. Put simply, it serves, served, or will serve a purpose. If this weren't the case, and spontaneous construction were considered design, then any physical process between objects would be design (this is a problem discussed later in this section). This purpose however, is necessarily dependent on the designer. For example, ants may construct a mound as shelter whereas humans may construct a mound to bury their dead. Furthermore, what the purpose is, and whether or not it even exists, is necessarily determined and perceived through human lenses. So then, similar to the problem with abstractly defining intelligence, a definition of design which encompasses agents a human would consider capable of design is necessarily framed by the human experience and therefore not independent of the nature of an agent. For example, could we say that ocean waves design the shores to be sandy? Well the ocean certainly does this, but does it have an abstract motivation or is the motivation purely physical? That is, is it simply "motivated" by the principle of erosion? This strongly echoes my previous points on intelligence in that it seems for a definition of design to exist independent of the human experience, it must contain a nonphysical element which influences motivation and intent (once again, free will is an example). Otherwise the motivation of any design is purely a result of physical laws (or in the language of the previous section, every output by an agent which impacts other objects, is a design on that object motivated by the input and rule set of the agent), and it would be impossible to introduce a standard independent of the human experience to differentiate what is or isn't a design. And just as before, this is not measurable and therefore can’t be known to exist within the agents humans typically consider capable of design.
Therefore, a definition of design which encompasses the agents humans typically consider capable of design, but is abstract enough to be independent of an agent's particular nature, is impossible to formulate without being meaningless. What humans recognize as design is necessarily dependent on the human experience.
Assuming we have an abstract theory of intelligence and design, would it be possible in principle to establish that some concrete object shows evidence of design by some intelligent agent (in this general abstract sense), solely by studying the properties of said object?
In order for an abstract theory of intelligence and design to exist, it must include a nonphysical element and not necessarily encompass agents a human-centric theory would encompass, as aforementioned. This nonphysical element, however, can not be measured. Therefore, no, it is not possible to establish that an object shows evidence of design by an intelligent agent solely by studying its properties.
Nature of the question
As I'm sure you've noticed, there's a certain pattern which emerged in the above passages which seems to indicate a deeper reason as to why the ideas you proposed are not possible. The essential problem lies in developing an abstraction from a particular agent's perception in order to describe any conceivable, or inconceivable, agent. The examples you gave are certainly intelligent, but only necessarily with respect to the human conception of intelligence. Any attempt to abstract this will be inherently human-centric, and therefore is not a true abstraction, or as you put it, would not be independent of "the exact nature or identity of the agent". To explain it from a different approach, the goal is to take what humans describe with a concept, and empirically construct a definition of the concept which can describe things nonhuman without human bias. The foundation of empiricism, however, is that our conclusions are solely dependent on our perception, and our perception is uniquely and necessarily shaped by the human experience. Furthermore, the concepts themselves are very human centric simply in their etymology. Intelligence and design are valued by human societies in that our ability to theorize and construct tools is very important to our survival. Having a different social and evolutionary framework of values, as other animals or abstract agents would, is something essentially inconceivable to us, particularly when considering frameworks which we don't even observe physically. Say we did have the grounds to claim a certain agent isn't very intelligent or is perhaps incapable of efficient design, and we may then cast judgement on them in a certain way for that. In an abstract space, that likely doesn't mean much (it depends on how the space itself is defined). The agent may not value those things and could equally accuse us of lacking X and being poor at Y, which it considers very important qualities. And while we may have a theory of the universe founded in intelligent design (e.g. many religions), this abstract agent may have a theory founded in XY, which is fundamentally different to ours or possibly even inconceivable to us. There would be no objective grounds however by which to judge or invalidate the other theory, for either agent. This is in fact a common argument against objective morality, in that any morality that humans construct can't actually be objective and is necessarily human centric, and the claim should instead be for a morality which is inherent to all humans (although this is of course an entirely different topic). So, in a similar fashion, intelligence and design are concepts whose meaning and inflated value have been ingrained into the human experience (whether by evolution, culture, god, etc.). To then try and define them in such a way which still satisfies how humans understand them but can also be applied to nonhuman, or even abstract, agents is impossible if not unfair. This definition will always favor intelligence which looks like human intelligence and design which looks like human design. More so, in the grand scheme of things, it simply doesn't make sense to arbitrarily take values or qualities out of the framework they were constructed in, and apply them to another agent which simply does not have the same framework. So sure, perhaps we can take an object and say that it has evidence of design by an intelligent agent, but only if that intelligence and design is close enough to what humans recognize as such. If there is indeed some cosmic definition for the concepts, it's useless to us because we can never escape our human bubble to see it, much less construct it ourselves. Again, if we theorize that an agent intelligently designed the universe, it's very much a meaningless statement because we can't know what intelligence nor design would look like for this agent, and if it is meant in the human sense of the words, then it is an entirely unjustified and human centric assumption to make about the character of this agent as well as the universe itself (I mean this in the purely analytical sense. Religious beliefs and faiths are fundamentally different, and I am not expressing any agreement or disagreement with them).
In short, the human experience is inescapable, and as such, any attempt to abstractly define intelligence and design would be necessarily human centric and how well we recognize that of another agent would depend solely on how closely that agent's expression of it is to the human conception. There is no meaningful objectivity which can be claimed, such as with abstractions in logic or linguistics (although, I believe this is even questioned). For us to even care about an agent's expression of intelligence and design is incredibly human centric, and can only tell us about the agent's relation to us, but nothing of it's nature or motivations.
In conclusion, a theory of intelligence and design is either human centric or meaningless. If a nonphysical element is introduced, then the theory may be either human centric or functional in abstraction, but meaningless in reality and therefore can't be applied to physical objects. Furthermore, the general idea of taking values and concepts that are inherently and entirely defined by the human experience and applying them to fundamentally nonhuman agents is deeply flawed.
*note: A key assumption throughout this argument is that physical laws exist and are equivalent in all frames of reference. If this weren't true then the physical laws would be objectively indeterminate, depend solely on the frame of reference (i.e. they'd be subjective), and it would be impossible to show that two frames have an equivalent set of laws. Then, any given frame could have a bias in terms of the rules or the complexity of the rules which an agent follows, allowing for a differentiation between what is or isn't intelligence/design. If this were the case however, then, given the reasoning in the above argument, an abstract theory of intelligence and design would also depend on the frame and there would be no way to show equivalence between frames, meaning it would be impossible to justify a consensus for any given theory. This would imply whether or not the properties of an object give evidence to design by an intelligent agent is entirely dependent on the frame, and a consensus or the setting of any standard would be unjustifiable. In this case, the answer to your question would be yes, but the results of it would be useless.