John Jay wrote to George Washington presiding over the constitutional convention to add the natural born citizen requirement to prevent a foreigner from being chief of the Army and the change was accepted without much discussion.
So the original intent of many of the founders was to let any long-term citizen be president. The requirement that the president be born a citizen was to prevent foreigners from becoming president.
So can Cruz be both born a citizen and born a foreigner?
Well, clearly he was. So, given the documented desire of tge Founders to preclude foreign influence from the office of Commander-in-Chief, it appears that the word "natural" actually does convey meaning above and beyond "born citizen." Hmmm, now what could that be, lol?