No it doesn’t mean that. It means you were a citizen at birth. You’re either an immigrant or a natural-born citizen. There’s no other status.
The recently deceased King of Thailand was born in Cambridge MA. He was eligible in your world.
So is every anchor baby and if one is only an American on one’s mother’s side like Ted Cruz, so was Winston Churchill.
Curiously Congress bestowed honorary citizenship upon him.
Ted Cruz wasn’t a citizen until he or his parents filed the correct paperwork to be recognized as such. When was that?
The poster onetime thought that a senator would have to be a natural born citizen.