No you misunderstand my original point. The child of American citizen(s) born anywhere is a natural born American citizen at birth. On that point we are in agreement.
My point was that natural born necessitated having allegiance to the country at birth. Children born to citizens of other countries are citizens of their fathers/parents country at birth no matter where they are born. It has been reasoned that while they are American citizens at birth since they are born here because there is the possibility of divided loyalties they are not natural born citizens. They are after all of dual citizenship at birth and are being raised by parent presumably loyal to another country, since the parents are not American citizens. This distinction does not appear to have held up over the years but it was that point that I was originally making.
I haven't read anything that supports that claim.
Then by that logic Ted Cruz cannot be a natural born citizen, regardless of his mother's nationality, because the circumstances of his birth gave him dual citizenship.