Sex is assigned according to seven different biological characteristics, which can and do vary considerably. Intersex children are assigned a sex depending on how closely these seven characteristics match male or female. A lot of people have six out of seven matching a particular sex, or five out of seven. A lot of people never find out that they have less than seven matching characteristics. However, this massively calls into question that there is such a thing as two distinct biological sexes.
Furthermore, why do we attach so much importance to sex (clearly a social construct) that we are prepared, as a society, to surgically modify babies?
Gender is how you feel on the inside. You can be non-binary, genderqueer, male, female, femme, butch, gender-fluid, etc.
A transgender person is one who was assigned a sex and a gender role at birth (in the arbitrary manner described above) but feels they are a different gender to the assigned one.
A cisgender person is one who was assigned a sex and a gender role at birth (in the arbitrary manner described above) but feels they are the same gender as the assigned one.
A genderqueer person is a person who feels like a mixture of genders. A genderfluid person is one who feels more like one gender on one day, more like another gender on another day. A non-binary person is one who doesn’t experience themselves as either of the binary genders.
Given the fact that the whole notion of a sexual binary breaks down upon closer inspection, and the notion of biological sex breaks down upon closer inspection, we are left with gender as the primary means of expressing one’s sense of self.
Therefore, if someone says they are a woman, they are a woman. Biological sex is a social construct.
And notions of third gender from other cultures don’t help. We didn’t grow up in those cultures, so we shouldn’t appropriate their gender constructs (which are part of a wider complex of social and cultural ideas).
Trans women are women.
Trans men are men.
Genderqueer, genderfluid, and nonbinary people are genderqueer, genderfluid, and nonbinary.