Suppose we briefly revisit Quine's famous explication (as he calls it) of 'a exists,' to wit:
1. a exists =df (Ex)(x = a).
In something more like English, an arbitrary individual a exists if and only if there exists something to which it is identical. [... W]hat it boils down to is that a exists iff a is identical to something that exists. And that makes for a circle the diameter of which is embarrassingly short.
I may be getting in over my head here, but from my (admittedly limited) understanding of Quine, I thought a major motivation for his position here was to set an explicit standard for defining a theory's ontological commitments; that is, to clarify when we are committed to something's existence, rather than just analyzing what existence itself happens to consist of. At least, the former is the main focus of his essay: 'On What There Is' (in From a Logical Point of View).
Quine starts off with "the old Platonic riddle of nonbeing" (p.1):
I cannot admit that there are some things which McX countenances and I do not, for in admitting that there are such things I should be contradicting my own rejection of them.
If we were to claim "Pegasus does not exist", McX might insist that the word 'Pegasus' must refer to something for our claim to have any meaning. Quine instead recommends that we follow Russell in translating apparent names into descriptions instead. For example, 'The author of Waverley was a poet' can be paraphrased into the nameless: 'Something wrote Waverley and was a poet and nothing else wrote Waverley'. From this we see that the original part 'The author of Waverley' does not demand any objective reference after all - this burden is instead carried by variables of quantification.
The virtue of this analysis is that the seeming name, a descriptive phrase, is paraphrased in context as a so-called incomplete symbol. No unified expression is offered as an analysis of the descriptive phrase, but the statement as a whole [...] still gets its full quota of meaning - whether true or false. (p.6)
So, saying "Pegasus does not exist" is no contradiction after all. We can paraphrase it as (say) "Everything fails to have the attribute of being Pegasus". (This may seem to commit us to an attribute of 'being Pegasus', but it certainly doesn't commit us to Pegasus itself.)
Quine is interested in how what we say gives rise to ontological commitments. So far we've seen that descriptions or alleged names can be squeezed out of. So what does commit us? That's where "to be is to be the value of a variable" comes in:
We can very easily involve ourselves in ontological commitments by saying, for example, that there is something (bound variable) which red houses and sunsets have in common; [...] But this is, essentially, the only way we can involve ourselves in ontological commitments: by our use of bound variables. [...]
We may say, for example, that some dogs are white and not thereby commit ourselves to recognizing either doghood or whiteness as entities. 'Some dogs are white' says that some things that are dogs are white; and, in order that this statement be true, the things over which the bound variable 'something' ranges must include some white dogs, but need not include doghood or whiteness. (pp.12-13)
So while Quine's definition of existence doesn't say much about existence itself, it does seem worthwhile for providing a clear standard against which we can assess ontological commitments.
Update: The following quote helps clarify things:
Now how are we to adjudicate among rival ontologies? Certainly the answer is not provided by the semantical formula "To be is to be the value of a variable"; this formula serves, conversely, in testing the conformity of a given remark or doctrine to a prior ontological standard. We look to bound variables in connection with ontology not in order to know what there is, but in order to know what a given remark or doctrine, ours or someone else's, says there is; and this much is quite properly a question involving language. But what there is is another question. (pp.15-16)