Senator Cruz’s father became a naturalized U.S. citizen in 2005. There is no distinction in law between a Citizen of the United States At Birth and a Natural Born Citizen.
Here is the relevant section of current US law:
8 U.S.C. § 1401
The following shall be nationals and citizens of the United States at birth:
(g) a person born outside the geographical limits of the United States and its outlying possessions of parents one of whom is an alien, and the other a citizen of the United States who, prior to the birth of such person, was physically present in the United States or its outlying possessions for a period or periods totaling not less than five years, at least two of which were after attaining the age of fourteen years”
There is a distinction up front in the Constitution as to ‘natural born citizen’ citizen for POTUSA and ‘citizen’ for Congresspersons. My brother, killed in the battle for Okinawa, and I, served in WWII on Leyte, were born as ‘citizens’ of the US in the US. Parents, were not citizens of the US. I never through years of reviewing my status had any reason to believe that I or my brother were “natural born’ citizens, instead just plain ordinary citizens. The code words you address notes specifically just plain ‘citizens’. There is a distinction contrary to your claim.