Well good to know
you did ignore my thing about people who have both ovaries and testes though

The definition of men used for klinefelter’s is “has a y chromosome”.
also when talking about sexes there’s two aspects, yes there’s gametes but there is anatomy, secondary characteristics, and so on. If we reduce sex to the gametes, then someone with ovaries but otherwise male anatomy is female, and somebody with underdeveloped ovotestes has no sex at all bc no gametes, and someone with ovarie(s) and teste(s) is both, so there’s atleast four if you count combinations. Then add in anatomy/organ/secodnary characteristsics mismatched with the gametes, and you have many more. Its easiest to lump all atypical sex into one and call it ”intersex” and say it is an actual sex, the only place where that doesnt matter is a sperm/egg bank XD