Gender is what societies assign to sex. The confusion over trans people is caused by people on both sides who conflate the two ideas. A trans woman is a social construct for a man who wants to live in ways that society usually assigns to women. You do agree that the idea of what's appropriate for men and women varies enormously from society to society, I hope. That's gender.

Gender theory is historically new. Before 1955 or so, people usually talked about sex roles, not gender.

To use your language, gender is the social manifestation of sex.