If you are a citizen of the US you are either natural born or naturalized, there are no other types of citizens.
uh no natural born means you have 1 way to determine citizenship and no claim to other countries. Your parents are US citizens and your are born a natural US citizen. If your parents are foreigners you can claim another citizenship other than US. So you are not “natural born.” You are just a natural citizen born in the US. That is why you see the different terms used because there are three ways to be a citizen. Naturalized is the third.
The founders didn’t want split allegiances in the position of commander in chief. How would you think they would want the son of a british citizen to be president after the two wars with Britain. The thought of it is ridiculous.