That's not the way it was taught when I was in school. As weird and stupid as it is, "natural born" was taken to mean born in the US. Period.
Then George Meany came along and that was resolved by saying that he was born on US soil in Mexico because his father was ambassador, or something like that.
Then McCain came along and he wasn't born on US soil in Panama, so it was further expanded - by new law, iirc - to include those born to US citizens serving abroad in the military.
And that's where it stands now. But the constitution says none of that, and until Meany and McCain "natural born" meant actually born in the US.
Don't get me wrong. I'm perfectly fine with those adjustments because, as I've repeatedly said, that restriction is dumb.
Personally, I see no reason why a naturalized citizen shouldn't be able to run.