The topic isn’t “Citizenship” it is “Natural Born Citizenship”. You will note that Article II uses TWO different terms. It uses the term “Citizen” and it also uses the term “Natural Born Citizen.”
If they meant exactly the same thing, they would not have included the specific words “Natural Born Citizen.” They would have simply used the term “Citizen.”
(Watch CodeToad whine about context from this excerpt I use to illustrate my point. :) )
“No person except a NATURAL BORN CITIZEN, or a CITIZEN of the United States...”
See? They are two different things. Article II SAYS so.
You are both correct and mistaken. You are correct in that there are two terms used. You are incorrect in what you think “natural born” means. What the clause says that either a person be born here (natural born citizen), or a citizen at the time the Constitution was adopted (citizen) is eligible to be President. That means that they considered someone that had already sought citizenship was a person whose loyalty was not reasonably questionable.