Oh dear, you don’t even know what a logical fallacy is.
“Someone says they don’t exist” is presented as a logical contradiction, not a logical fallacy, though on closer inspection it is not even a logical contradiction. A logical fallacy means the logical structure has a flaw, a logical contradiction just means two terms cannot be both true given a logical structure that isn’t flawed.
Invert your example to “Someone says they do exist”, and that is a logical fallacy: Begging the Question - the conclusion is already assumed in the wording. One can therefore logically conclude (since “exist or don’t exist” covers all possibilities exhaustively), that “Someone says they don’t exist” doesn’t contain a logical fallacy. Either that or the statement itself is invalid because there is either a semantic or syntactic flaw with the subject and/or predicate.
If your example were to contain the “Tu quoque” fallacy like you’re trying to pull, then out of the two options available, that leaves us only with a semantic and/or syntactic flaw with the subject and/or predicate. If this is the case, then we might look at how your example suggests a paradox similar to the liar’s paradox: if someone says it’s true that they don’t exist, they don’t exist to say they don’t exist, but through the act of saying, they demonstrate existence, and you go round and round. However upon closer inspection it’s more like Zeno’s paradox, which turns out not to be a paradox if you simply unpack it. Unpacking it reveals, as I keep saying, the problematic notion of the “identity” of the “someone”. Now it becomes clear that there is an issue of the False Dilemma fallacy in presenting only two options: either “someone existing” or “someone not existing”. The third option being that there is existence that contains within it what is incorrectly isolated “identity” of the “someone”. Now the congruence of the sentence itself is exposed as flawed - and not the implications of the sentence, given its assumed coherence.
Either that or the example doesn’t contain the “Tu quoque” fallacy afterall - take your pick. But this is all stuff I’ve already brought up, I saved the best 'til last:
The hilarious thing is that not only do you not understand what a logical fallacy is, you don’t seem to realise you would be committing the “Tu quoque” fallacy yet again if you were implying that the problem with “Someone says they don’t exist” being because their actions say they exist even if their words say they don’t, is that someone is not acting on what they are saying - and therefore what they’re saying is wrong. You’re trying to offer “a basic counter example of a so called logical fallacy” by committing the very same logical fallacy, as proof that logical fallacies can be countered?! So even on what you call “basics” you fail.
Basically on all sides, everything you say demonstrates that you are a newcomer to logic and you are pretending you can swim way out of your depth.
The reason you don’t think I’m as smart as I think I am is because you’re not smart enough to realise I am. If you were, you wouldn’t make that remark - how is someone supposed to recognise what smarter looks like if they’re not smart enough to know what it looks like?
You need to learn some humility as soon as you can - learn to admit where you are, err on the side of caution until you’re sure you’re where you say you are. It’s great that you want to fill boots that are way too big for you, and I commend your new interest in logic - it will help with your ambitions, the more you learn about it. But to get there, put aside your pride, ego, and premature declarations of your brilliance until you know what you’re talking about and you can legitimately back it up - we’ve all been beginners once so there’s no shame. Good luck.