It works the other way, of course. But if a couple from Waco are vacationing in Cabo and the wife goes into labor, does the child have Mexican citizenship through “place of birth?”
Yes, and the parents will need to register the baby in Mexico and get a Mexican birth certificate for the baby before the U.S. consulate will issue a Report of Birth Abroad (essentially equivalent to a U.S. birth certificate).
What happens if parents are of different citizenships? Like, Mexican marries American, moves to the States but hasn’t yet gone for official US citizenship. Baby born in the States. Does the kid have any claim to Mexican citizenship? Is the answer the same for people from all different countries living in the States as legal aliens, or is that the sort of thing determined by the laws of the country of origin?
The child of a Mexican citizen who is born in the United States is entitled to claim Mexican nationality and citizenship, which may be certified at the Mexican consulate corresponding to the state where the child was born. In this case, the parents would have to provide the U.S. birth certificate as proof of birth and parenthood.
An interesting bit of historical trivia: My great, great-grandfather holds the distinction of being the first person born in Mexico (if you discount the fact the Texas was originally Mexico) to serve in the Texas legislature. His father was in Zach Taylor’s army in Monterrey and his mother (and many other wives) accompanied the army. His father died of cholera shortly after he was born and his mother returned to New Orleans, remarried, and moved to Texas.
I doubt that his parents registered with the Mexican authorities, seeing how there was a war going on at the time.