You're are correct with regards to Congress having the enumerated powers with regards to the naturalization process. Which is quite different from being regarded as a "natural born" citizen. Naturalization can take place either by an immigrant coming here and going through a process to obtain citizenship, but it also can be bestowed upon persons at birth. A "natural born" citizen needs no confirmation that their allegiance is limited to the United States and the United Sates alone. That would be a person born within the territorial confines of the United States to citizen parents (plural).
In addition to the Constitutional requirements stating only a "natural born" citizen who has obtained the age of 35 is qualified to become President of the United States, the following must also be taken into consideration when determining "natural born" citizenship.
The United States vs. Wong Kim Ark.: "A person born out of the jurisdiction of the United States can only become a citizen by being naturalized...."
U.S. Department of State Foreign Affairs Manual: "Automatic" acquisition of U.S. citizenship after birth, a form of naturalization by certain children born abroad to U.S. citizen parents or children adopted abroad by U.S. citizen parents.
Ted was born abroad but to only 1 citizen parent. However, even if both had been citizen parents, he would still be considered a naturalized citizen
That is where we disagree.
You were correct on most of what you stated, but Cruz is not a naturalized citizen. US Code provides for two methods of citizenship, by birth and naturalization. Cruz obtained derivative citizenship (jus sanguinis) thru his mother. This is considered by birth, since Cruz did not go thru the naturalization process, which is distinctly different than derivative citizenship both in process and substance.
However, the issue of NBC is an open question and needs to be resolved thru the courts. It is not settled law.