You actually just proved what’s probably the opposite case you intended to prove. The Natural-Born Citizen movement has been trying to prove that when Vattel’s use of the term “les naturels” should be regarded as the basis for the concept of “natural-born citizenship.” Vattel went at very great length to define “les naturels” as the natives of a country.
What you’ve done is demonstrated that, as a point of law, “native” is not identical to “natural-born citizen.” Who else are natural-born citizens, besides those who are native? Those whose parents are citizens of a nation! So, yes, Ted Cruz and Marco Rubio qualify as natural-born citizens.
Natural-born citizen means those who are citizens as a legal consequence of their birth, as opposed to those who are naturalized. Most NBCs are thus because of where they are born (”native”) but not all.
Other people who are NBC, but not natives:
John McCain (who was born in Panama, of U.S. citizens) and Bobby Jindal (who was born under U.S. law, but whose parents were not Americans)