What is the difference between a citizen and a natural born citizen?
Clearly from the language of Article II Sec.1 there is a difference.
The difference is that one can only naturally be a citizen when one cannot be anything else.
The children of foreigners are born something else from their foreign parent(s) and are therefore not naturally Americans.
Same principle that gives Ted Cruz any claim to US citizenship being born in Canada, he inherited it from his mother.
He is just as eligible as Anwar al-Awlaki’s kids, born in Yemen to one citizen parent.
Or all the Chinese tourist babies.
Your initial question, I presume, pertains to the USA in scope, and attempts to get at the difference between “naturalized” and “ natural born” citizen, with an eye toward qualifications for the office of POTUS. Most who seek public office do not aim so high. How do we identify the pretenders by way of inquiry?