a hallucination is in a sense a lie.
which can easily be perceived and experienced, regardless of the fact that they are untrue.
so they're not lying about having a hallucination, but if they said "dude look at that ice cream, it's talking to me", and they were hallucinating, then the statement is not true.
the idea of existing is something that is real, and true, as opposed to something that is a lie and unreal, such as an illusion or a hallucination.
if you need me to be super specific as to what I'm talking about, to leave no possible loopholes, then I will be.
Take existing to mean something true, and real.
In reference to a person, existing means that they are in fact a living, breathing, thinking, judging, etc... person.
so, 'prove to me that you exist' asks for proof that the person in question is indeed a living, breathing, thinking, judging, etc... person, and not in some way a fake person or part of an illusion or hallucination.
stop trying to redefine the question as I intended it, or your answers will not fit the question I posed.
And yeah, I'm bad at letting things go too.