Yes, but debates over the meaning or usage of words are social/philosophical/political issues, not biological issues.
> Many people don't accept that belief, for a variety of reasons, such as viewing that belief as being based upon sexist stereotyping of women and men.
This claim seems questionable. As far as I can tell, the origin of the opposition is mainly religious, and it has come to be political as a consequence of the religious leanings of political parties.
> Instead, they understand "woman" and "man" to be the words used to describe, respectively, female and male people who have reached adulthood.
Yes, but it's unclear how a study of brain matter would change anyone's mind. After all, their definition of female and male always depended on genitals rather than brain composition.