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.
This is Ted's situation.