Since a child born to American citizens is considered unequivically American and always have been as far as I know with no constraints the clause clearly was intended to apply to naturalized citizens IMO. I think it's pretty clear there were concerns about foreign puppets in the early years. If it was meant to allow those with only the most unquestionable allegience by location as well as heritage I'd think the phrase would have been something like natural born and raised (and possibly educated). I find it hard to imagine they would have intended to penalize the descendents of soldiers defending or fighting for their country over seas.

Heck if there's any doubt about the obvious meaning maybe Arnold can claim he's a natural born American by his obvious personality fit with the American system through his business and political success - like a natural born swimmer, runner, lover etc.

Regards,
Mark