Sex is a spectrum, and is not binary, just like gender. Intersex people also exist, and gender is caused by genetic traits and development, so everything has to be taken into account, you can't just go straight to a single factor to determine everything. You also have to look at, what about a cis person getting bottom surgery? That has happened before because of errors during medical procedures, so what would they be?

And legal gender/sex markers are usually the same, but some places have them separate. For example, I'm from a part of the US where we can submit a form and get our birth certificates changed. That is considered changing your legal sex. In some other states, you can't really change your legal sex, but places will let you change your legal gender marker on certain documents, basically what shows you as m/f/x on passports and drivers licenses.

There truly is no easy way to define what a "woman" is. Same thing with a man, always remember trans men and nonbinary people exist and are just as common as trans girls.