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I'm a little unclear on what your criticism is. Is one of these right?
You're being too precise, whereas I wanted to have an informal discussion in terms of our everyday intuitions. So definitions are counterproductive; a little unclarity in what we mean is actually helpful for this topic.
There are two kinds of existence, one that holds for Plato's Realm Of Invisible Mathy Things and one that holds for The Physical World. Your definitions may be true of the Mathy Things, but they aren't true of things like apples and bumblebees. So you're committing a category error.
I wanted you to give me a really rich, interesting explanation of what 'existence' is, in more fundamental terms. But instead you just copy-pasted a bland uninformative Standard Mathematical Logician Answer from some old textbook. That makes me sad. Please be more interesting next time.
If your point was 1, I'll want to hear more. If it was 3, then my apologies! If it was 2, then I'll have to disagree until I hear some argument as to why I should believe in these invisible eternal number-like things that exist in their own unique number-like-thing-specific way. (And what it would mean to believe in them!)
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.