I’ll keep it simple: since the ratification of the 14th Amendment, there are two and only two categories of US citizenship: born citizenship and naturalized citizenship.
Born citizenship can be acquired either by inheritance: “jus sanguinis” or by being born in the U.S., “jus soli.”
Born citizens can be president; naturalized citizens cannot be president.
You’ve oversimplified. There were the same two kinds of citizenship before the 14th amendment too. The only difference is that fewer people were considered stateless or aliens after the 14th amendment was passed. It didn’t redefine natural born citizens ... and not all citizens at birth can be president.