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How about this: Mathematicians have a conception of existence which is good enough for doing mathematics, but isn't necessary correct. When you give a mathematical definition of existence, you are implicitly assuming a certain mathematical framework without justifying it. I think you would consider this criticism to be a variant of #2.
In particular, I also think about things mathematically, but when I do so, I don't use first-order logic, but rather intuitionistic type theory. Can you give a definition for existence which would satisfy me?
I'm a mathematical fictionalist, so I'm happy to grant that there's a good sense in which mathematical discourse isn't strictly true, and doesn't need to be.
Are you asking for a definition of an intuitionistic 'exists' predicate, or for the intuitionistic existential quantifier?
(Note: I added a link in my previous comment)
First, if you accept that mathematical constructs are fictional, why do you consider it valid to define a concept in terms of them? Second, I admit I wasn't clear on this issue: The salient part of intuitionistic type theory isn't intuitionism, but rather that it is a structural theory. This means that statements of the form "exists x, P(x)" are not well defined, but rather only statements of the form "exists x in A, P(x)" can be made.