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The words "man" and "woman", in their most modern usage, seem to be a map-territory confusion.
Defining these words intensionally (in terms of other words) usually ends up circular. Defining these words extensionally (by pointing at examples) seems to give us a pretty simple intensional definition: "A (woman|man) is someone with a deeply felt sense that they are (female sex|male sex)."
Since people consider their brains to be their "real" selves, they may also think their internal sense of their sex is their "real" sex. This isn't an explicit belief anywhere, but it shows up in the way people talk about it.
But your sense of your sex is not your sex. It's a reference to your sex, and it's part of the map. If your sense of your sex (your gender) doesn't match your sex (your body), your sense of your sex is wrong. Maps that don't match the territory are wrong.
Moving the word for "adult human females" to instead mean "Everybody with a felt sense that they're adult human females" sets up a motte and bailey. The motte is "Gender is separate from sex," and the bailey is "Trans people's felt sense of their sex isn't wrong."