I would strike the word “traditionally” from that definition since we’re talking about a comparative and differential analysis and concentrate on the “qualities and characteristics” part. Although most people in the US today wouldn’t think of it this way, imagine the perception of a woman army officer commanding male troops in 1845. You can take the same approach when looking through history or across cultures. What roles, qualities, and characteristics are associated with “women” and how do they differ and evolve?

There’s some complexity when you get into the details - indigenous cultures change when they come into contact with, say, colonialism, and the people who studied them might themselves be observing through their own prejudices. History is replete with examples of British colonialists being unable to properly deal with things like the egalitarian democracies of the northern indigenous peoples or the matriarchal social structures. Picture the used car dealership where the salesman still insists on engaging with the man even though it’s the woman buying the car.

Semantics is the study of the meaning of words, and semiotics is the study of symbology. When we’re talking about these things, we’re talking about how the ideas and symbols associated with the idea-token “woman” differ.

The reason why this is important is that this is the crux of the transphobic argument. Their argument is cultural, not biological (although like I said, even their biology is sketchy).

I think a great study that includes cross cultural anthropological analysis of the role of women, as well as politics and economics, is David Graeber’s The Dawn of Everything.