Okay, this is all true. However, biological sex is not exactly the same as gender. Biological sex is fixed. It cannot be altered. You have defined it exactly correctly. However, gender roles can vary based on time and place. There have always been some people who don't fit the traditional gender roles of their societies, biological sex notwithstanding. For example, I think things like being more social or domestic are thought of as feminine characteristics, but they are more cultural than biogical.