“And many Presidents and Vice-Presidents have been born in territories, the District of Columbia, and even in foreign countries (the children of US citizens are natural born citizens even if born in a foreign country).”
A natural born citizen is born in a nation of citizen parents.
Naturalized, derived, and native-born citizens can all serve in the House of Representatives and the Senate.
The Presidency requires a higher form of national security inherent in native born plus derived citizenship (jus solis and jus sanguinis).
Barack’s father was never a US Citizen. And his mother was a minor at time of birth if I remember correctly.