What you're describing about gender is.... not really scientific. It was basically declared by fiat by researchers. It's not an authoritative definition and many people disagree with the concept, at least when it gets conflated with scientific topics.

there are some things that clearly exist that are really hard to nail down with definitions; once we get into anything social, we are kinda playing with that territory. everything is so fuzzy that our normal way of defining things breaks down. so saying precisely what gender _is_ is going to be almost impossible. but there are definitely roles and traits that are highly correlated with a person's birth sex that are distinct from their birth sex. there can even be genetic reasons why those correlations emerged. but they are still distinct.

as evidence, what it _means_ to be a man, woman, etc, differs from society to society. if you ask me to quantify this precisely, i will struggle. but it's plain for all to see.