“So an anchor baby can run for president?”
I believe the parents must be intentionally subject to the law of the United States at the time of the birth (e.g., green card residents or at least a visa vs. illegal aliens and/or children of diplomats).
Natural born includes those born of American parents on American ships in foreign ports, or on the high seas, or in American Embassies and Consulates, all of which are considered by the principles of ex-territoriality, as within the United States and under its laws.
Children born to American parents traveling in foreign countries are regarded as native born, instead of natural born.