Is there a suppressed premise in Anselm's Ontological Argument?

Question

Source: 5 minutes 20 seconds juncture; Lecture 1, Video 4 (transcription);
MITx: 24.00x Introduction to Philosophy; by MIT Associate Prof Caspar Hare PhD (Princeton)

[...] Suppose God existed in the understanding, but not in reality. Here He is-- God, there in the understanding but not in reality. How great would He be? Well, pretty great. After all, we're thinking about God. It's a pretty great thing.

But now suppose that God existed in the understanding and in reality. How great would God be then? God would then be extraordinary. After all, God, if He existed in the understanding and in reality, would have created us all, would have truly true omnipotence, would have made the universe around us, would be interfering-- well, how much he interferes, of course, depends on your theology-- but would be the kind of creature who we should think of as really, really, really great. God would be better if He existed in the understanding and in reality than if He just existed in the understanding.

But what does that mean? It means that God can't exist in the understanding, because remember, Anselm said it's part of our idea of God that we can't imagine Him being any better than He actually is. If you're thinking of something that could be better than it is, then you're not thinking of God. So for example, if a fool says, I'm thinking of God as existing in the understanding but not in reality, then the fool is not thinking of God. God can't be in the understanding but not in reality.

But God, of course, is in the understanding. The fool claims to be thinking of God. We can think of God. It's easy to think of God. We just think of a being such that no greater being can be imagined. And so God must be there in the understanding and in reality. He must be real.

Call the premise below (h). Notice that it doesn't appear in the above transcript.
Is h) cogent or not? Does (h) explain the bolded above? How does the bolded above follow?

[Source:] At this point Anselm wields what is perhaps his most controversial premise. It is hard to know exactly how to formulate it. But something like the following seems to be what Anselm has in mind.

(h) If something exists in the understanding alone, but can be conceived to exist in reality, then that thing can be conceived to be greater than it actually is.

The idea seems to be: if we compare two things that are alike in all respects, except that one exists in the understanding alone and the other exists in reality, then the one that exists in reality is clearly greater, better, more perfect.

Answer 1

Here is Plantinga's reconstruction of Anselm's ontological argument:

"(1) God exists in the understanding but not in reality. (assumption for reductio)

(2) Existence in reality is greater than existence in the understanding alone. (premise)

(3) A being having all of God's properties plus existence in reality can be conceived. (premise)

(4) A being having all of God's properties plus existence in reality is greater than God. (from 1 and 2)

(5) A being greater than God can be conceived. (3, 4)

(6) It is false that a being greater than God can be conceived. (by definition of 'God')

(7) Hence, it is false that God exists in the understanding but not in reality. (1-6 reductio ad absurdum)

And so, if God exists in the understanding, he also exists in reality; but clearly enough he does exist in the understanding (as even the fool will testify); accordingly, he exists in reality as well."

Your proposition h) is a conjunction of 4)-6), so it is cogent, but as you can see none of them is a premise.

The problem with the argument is in the premises 2),3). In Kant's critique of the variants of this argument due to Descartes and Leibniz, it is expressed as pointing out that "existence in reality" is not a real property, see What are the counterexamples to Kant's argument that existence is not a predicate? As such, it can not be added to other properties, and then conceived or compared to other conceptions. Here is Kant:

""Being" is obviously not a real predicate; that is, it is not a concept of something which could be added to the concept of a thing... By whatever and by however many predicates we may think a thing - even if we completely determine it - we do not make the least addition to the thing when we further declare that this thing is. Otherwise it would not be exactly the same thing that exists, but something more than we had thought in the concept: and we could not, therefore, say that the exact object of my concept exists."

In other words, on Kant's view premise 2) is plainly false. Schopenhauer was more acerbic:

"On some occasion or other someone excogitates a conception, composed out of all sorts of predicates, among which, however, he takes care to include the predicate actuality or existence, either openly or wrapped up for decency's sake in some other predicate, such as perfection, immensity, or something of the kind".

Plantinga tries to get Anselm out of trouble in the linked paper by using semantics of possible worlds to redefine what it means to be a property, so that "properties plus existence in reality" can still be conceived. However, this type of semantics is itself controversial, and it is unclear why one should accept it if one does not accept Anselm's premises. But if one does accept that "existence in reality" is a conceptual property it becomes unclear what it is for it to be "conceived" over and above the "existence in understanding", and why one should accept premise 3).