I just want to point out how problematic it is that you define gender by the ability to become pregnant. You say that anyone born with a vagina is a woman, yet many people born with vaginas do not have the ability to get pregnant or choose not to get pregnant. That does not mean they are not women.
That was not meant to be a "definition" of gender, but a clear example of the physical attributes that, together, have defined gender for as long as the concept has been around. From the Merriam-Webster Dictionary (non-relevant definitions removed):
GENDER
2 a : sex
b : the behavioral, cultural, or psychological traits typically associated with one sex
SEX
1: either of the two major forms of individuals that occur in many species and that are distinguished respectively as female or male especially on the basis of their reproductive organs and structures
2: the sum of the structural, functional, and behavioral characteristics of organisms that are involved in reproduction marked by the union of gametes and that distinguish males and females
3a : sexually motivated phenomena or behavior
b : sexual intercourse
The IF/THEN logic actually goes the other way around: If a person gets pregnant and carries/delivers a child, then she IS female. If a person produces semen and impregnates a woman, then he IS male. Lack of either of those actions/attributes does not disprove gender, but presence of one of those mutually exclusive attributes proves gender.
While there is a trend to disassociate gender with physical attributes (micro- as well as macro-), and to associate it instead with behavior, I personally don't buy into that theory. The concept of transgender, to me, is strictly a behavioral or role-playing concept associated with the "typically" verbiage in "GENDER" definition 2b above.
I do accept that homosexuality - sexual attraction to members of the same sex - is a reality that goes beyond chosen behavior. I doubt there is a direct link between homosexuality and transgender. Rather, I perceive transgender more as an extreme form of transvestitism, where the person seeks physical change to go along with the clothing and other trappings of "typical" sexual roles.