That's true, but since the Constitution defines very very few of its terms, we must look elsewhere for the definition understood by those who wrote and ratified the Constitution.
We know the term "natural born citizen" and the variant "natural born free citizen", was in use well before the Constitution was written. But almost every use gives no hint as to it's exact meaning. However the Journals of the Continental Congress, for July 27, 1781 documents a translation of the French "naturels" to "natural born" in a secret agreement with France. Vattel, in French, said that "naturels" and "indigenes" were those born in country of parents who were citizens. Many have argued that "naturels" means natives, and "indigenes" doesn't mean naturals or natural born either. (In reality depending on context, either word could be translated as "naturals". But apparently those who translated that 1781 treaty felt "naturales" when modifying "subjects" was equivalent to "natural born". If that was the understanding, then Vattels "naturels" could also be "natural born".
The evidence is quite strong that "naturales" was understood, in these sorts of contexts, to mean "natural born". Thus the case for the Vattel "definition", requiring birth in the country (with exceptions for military and diplomats) of citizen parents, being the one the founders understood for "natural born citizen", is very strong.
Another way to view this is writing what the founders could have said if they wanted Native born citizens to be president.
They could have easily stated that those born on US Soil are eligible. Instead, They chose the NBC term with “Citizen” in it . US Citizenship at birth is the definitive qualification. Obama was born with foreign citizenship.
In the Congressional record natural-born is hyphened. The first English edition(1797) of Vattel using natural-born is hyphened.
Connection?