I think the question setup is somewhat flawed but there's still a partial answer that can be offered in response to it. I think this paragraph has several problems:
But I think it is obvious that before asking "Does X exists?" one should provide an exact definition of "X", so I guess that those who engage in this nihilism/realism debate agree on the definition of "moral fact/property".
No Perfect Definitions are Necessary
First, it's far from obvious that we must always have a "exact definition of 'X'" before we can debate X's existence. It's easy to show why with a reductio.
(1) Exact definition = knowing exactly what something is = knowing everything about it. (guessing what you mean by that).
(2) If we cannot talk about something until we know everything about, then we can never talk about anything. (seems to be what you're implying about morality).
(3) Ergo, we can never talk about anything.
Or we must reject the argument. I suggest we reject both (1) and (2). While I'm having some trouble finding the quote, I remember quite clearly Aristotle saying something to the effect of their must be an end to definition at some point if we are to accomplish anything.
Another practical example, we discovered that Neptune exists before we knew anything directly about it. The only thing we knew is some object is exerting gravitational pull on other objects and messing with their orbits.
The Nihilist does not need to know the definition
Second, yes, the nihilist denies there is any such object, but this does not mean the nihilist knows what such an object would be like. Presumably, the nihilist believes that any such object is incoherent at its core or incompatible with other metaphysically important things.
Incoherent at its core = (perfectly) round square.
Incompatible with other features of metaphysics = an object that has no energy or mass (defies the definitions of energy and mass and how we understand them to work).
Presumably, the nihilist takes moral facts to be this sort of thing that cannot have a fundamental existence. Perhaps, they deny anything non-physical has existence (and assume that moral facts are not physical themselves). Perhaps, they deny anything cultural has real existence (and assume that moral facts are merely evolutionary feelings). But it's not greatly important to them precisely how moral facts are understood or defined, because clearly the cart comes before the horse here. Moral facts are denied for other logical reasons.
Moral realists do not need to specify the definition either
If wikipedia presents moral realism as a single view, then it does moral realists an injustice. Moral realism is a description of the metaphysics of several different views. Some might not greatly prefer having their claims articulated in terms of the reality of certain propositions, but what all views share is belief that moral statements inhere in the world.
They don't need to agree on the definition, because they don't have to share an account of how they inhere. So, it doesn't matter if someone is a natural law theorist or a traditional Kantian or a utilitarian (i.e., they don't have to agree about what is right/wrong or the good). What unites these views is that they each assert there is some real existing thing that is good or evil.
The skeptical possibility
You can choose to deny my first section. You can believe we should never take positions on anything we cannot define exactingly. This would make you a type of skeptic.
You could even pick to be a skeptic about morality alone. So you could accept scientific facts as real and provable but believe we cannot know whether moral assertions attach to the world in the same way. I don't think that's a very popular position, but there's nothing in logic alone that disproves it -- I just don't see how anyone can live that out.