There is also historical precedent that Natural Born Citizen means being born, and not naturalized, in the US.
The first presidential nominee of the Republican Party, in 1856, was John Charles Fremont. He was born in Georgia to an American mother and a French father. Jean Charles Fremon was born a French citizen, near Lyon, France. He was not a U.S. citizen at the time of his son’s birth and never did become a citizen. Abraham Lincoln campaigned for Fremont. All the founders of the Republican Party campaigned for Fremont.
So, we have strong historical precedent.
Well, Fremont never became President.
The only precedent we have is Chester Arthur, who lied about his father’s citizenship status at the time of Arthur’s birth.
His father was a Canadian citizen. The lie was the father had become a citizen of the Us prior to Arthur’s birth.
Quite a big deal was made of it at the time, and records were not like they are today, so he got away with it. Later on, the lie became known.