This distinction between sex and gender is a common use of those two terms (at least in particular contexts) but the idea that "man" and "woman" identify genders and "male" and "female" identify sexes is much less common. More commonly, "male" and "female" are adjectives pertaining either to sex or gender and "man" and "woman" are mind referring to adults who are, respectively, male and female, again, either by sex or gender.

That's just a question of words and their definitions.

Personally, I use both woman and female completely the same, and sometimes it means women and trans-women (e.g. when I see someone who looks typically female), but mostly it means born women only (e.g. when I say I'm attracted to women).

But of course, not everyone uses the same words with the same meaning. There's no need to assume bad faith.

Words and their definitions are pretty much what society is built on and it's useful to have words mean different things since it is what allows us to express so many different ideas -1984 famously destroyed the vocabulary of society expressly so that only the correct ideas could be considered.

Without a difference in meaning, why have different words? It seems preposterous to extend the English language just to play scrabble.

Right, but the original meaning of "woman" is "a female human", whereas "female" is an adjective or a noun for one of the biological sexes (the one that produces ova in sexual reproduction). Including trans-women in the definition of "woman" is a modernistic redefinition of the word, so people are totally justified if they use the word with its original meaning. Maybe we should make up a new word for the new meaning?

That might have been a sensible choice about twenty years ago. As it is the other usage seems to have been established too much now. This isn't anything new, words change meaning and tone all the time.

I think that only 10% of English speakers think trans-Women are actually women. However that 10% are disproportionately powerful, so they are able to control how the word is used in the media, in the workplace, and in educational institutions. If they stopped doing this, usage would revert to its original pretty much instantaneously.