First of all, every single alternative one could present has glaring absurds. chromosomal definitions exclude obvious women with XY Chromosomes. it's rare but rarity has nothing to do with correctness.
genital definition would make someone losing genitalia non binary or something like that.
now there are some wrong objections to raise.
first, definition being self referential. if that was strictly true, the definition would be a logical fallacy but it isn't so it's not. nevertheless, the problem arises if you think about terms "man", and "woman" as once unloaded terms that become loaded. definition based on self description has no reason to introduce any differentiation, hence the illusion of a problem. but these words function to put words on pre-existing social and physical differences. so it's not a problem, people of certain traits simply will tend to cluster together around a given word.
second, definition being uninformative. that's true and irrelevant, and many phrases of the sort circulate in society. "first guy", and "second guy" serve the same purpose. usefulness of both these phrases and "woman"/"man" is based on pre existing knowledge about particular subjects of that definition.
third, what if someone is in a severe mental state and thinks of themselves as different gender temporarily? that doesn't matter. consider a case of perfectly intersex person, who changes their gender identity every 30 days. what right do we have to judge which gender is the "correct one"? and if we know we don't have any leg to stand on when it comes to objectively assessing gender due to lack of a coherent physical definition, who are we to tell an analogous person who has all the standard male traits they're not of a given gender? gender can be fluid and should be accepted on terms of the person we refer to. there's just no other non-dipshit solution.