What definition of to exist, of to be, and of real preponderates contemporary philosophy? How do the terms differ from one another?
Most philosophers would hold that all three of those terms are synonymous. The seminal paper on this question is Quine's On What There Is. For Quine the only things that exist are physical objects.
However, this wasn't the view of some very important ancient philosophers like Aristotle and there are at least a very few contemporary philosophers who would hold that different kinds of thing exist in different ways. One of these folks is Kris McDaniels, who has a paper called Ways of Being or something like that.
In 'The Basic Problems of Phenomenology' Heidegger describes two types of being: the existence of things perceived is one type, and such things are extant beings; but there is also the existence of the perceiver, which is a qualitatively different type of being, (transcendental). Heidegger called this the ontological difference.
The quotes below are from Basic Problems, on the terminology of 'real' in Kant :-
The concept of reality and the real in Kant does not have the meaning most often intended nowadays when we speak of the reality of the external world or of epistemological realism. Reality is not equivalent to actuality, existence, or extantness. It is not identical with existence, although Kant indeed uses the concept "objective reality" identically with existence.
The Kantian meaning of the term "reality" is the one that is appropriate to the literal sense of the word. In one place Kant translates "reality" very ﬁttingly by "thingness,” "thing-determinateness." The real is what pertains to the res. When Kant talks about the omnitudo realitatis, the totality of all realities, he means not the whole of all beings actually extant but, just the reverse, the whole of all possible thing-determinations, the whole of all thing-contents or real-contents, essences, possible things. Accordingly, realitas is synonymous with Leibniz' term possibilitas, possibility. (Page 34) ...
The Kantian concept of objective reality, which is identical with actuality, must be distinguished from the concept of reality as thus elucidated. The realness or being-something that is fulfilled in the object thought in it, in its Objekt, is called objective reality (objektive realität]. That is to say, it is the reality exhibited in the experienced entity as an actual existent entity. In reference to objective reality and reality in general, Kant says: "As regards reality, we obviously cannot think it in concreto without calling experience to our aid. For reality can only relate to sensation as material of experience and is not concerned with the form of the relationship, whereas, if we so chose, this form could be made subject to a play of fictions." Kant here separates objective reality as actuality from possibility. (Page 37)
I find this give a nice conception of reality from the existential point of view, because one is dealing with possibilities. Scheodinger's cat is alive and dead in 'reality', until the actuality is ascertained.
Other philosophers no doubt use different terminology.
I don't know if this would really answer your question, but it should at least indicate where & what to look for :)
to be - in general form means the same as to exist (a special example as expressed by Descartes: Cogito ergo sum - popularly translated as 'I think therefor I am (have body)' <=> because 'something' needs to contain or do the thinking)
real - is the same as to exist. (To say something is real if it exists is tautologous.)
to exist: This becomes valid if and only if there is someone/thing that can be aware, otherwise the question would not exist(!). In other words it refers to perceptibility: what you are aware of or can infer from what you are aware of (including the phenomenon of awareness).
This changes your question to become: "What is awareness?"
They differ to the extent that the author using them fails to properly specify exactly what they are talking about. The issue of what these three terms actually refers to, I think is answered nicely thus:
According to the article series 'Highly Advanced Epistemology 101 for Beginners' by Eliezer Yudkowsky; we can recursively define the physical universe.
His work takes root in the idea that all complex systems can be understood by examining their constituent parts and applying the theory of emergent phenomena: Brains are collections of neurons, neurons are collections of organic molecules, organic molecules are collections of atoms, atoms are collections of subatomic particles, etc.
Then, he introduces the concept that mathematics is a separate thing from the computer or human reasoning about it; and the mathematical reasoning is in some sense a "window" into the world of maths. This is akin to platonism but subtly different: He describes maths as (paraphrased) "The fact that when you start out with a certain well-defined symbol string, and apply certain well defined transformations to it, you always end up with the same result." No mystic dimension, just a statement of an apparent fact about the universe.
He then goes on to argue that in fact, what happens when one reasons about mathematics, is that some physical system takes on the properties of a mathematical model of an axiomatized theory. Pebbles in bowls behave like Peano Arithmetic (with an upper bound), as do our neurons when we count, sum and multiply, and as do the transistors in a computer when we use it for calculation.
His central thesis is then that everything can be split up into parts which can be put in either "Physical universe" or "Mathematical object" boxes.
The definition he provides of the "Physical universe" is based on Judea Pearl's Causality (2000), which defines Causality in terms of conditional probabilities. That there is a great web of causal arrows, connecting all events in the physical universe, going from past to future events, to a greater or lesser extent.
Yudkowsky's definition of the Physical Universe is then that one starts with the phsyical process that implements one's own cognition (the brain), adding it to the set of "the physical universe". Then for every object in the "physical unvierse" one adds all objects causally influencing or being causally influence by said objects.
That is, everything ever influenced by or influencing anything influencing or being influenced by you, in any in principle observationally determinable way is 'physical' and 'exists in the real world'. Everything else is mathematics.