Are there more than two types of American citizens, Natural Born and Naturalized? If not, then aren’t you Natural Born if you are an American citizen at birth?
An excellent and succinct question.
At this point the de facto consensus is that there are two types of citizens: citizens at birth, and naturalized citizens, as you suggest. Thus the "Natural Born Citizen" term used in the Constitution is taken as a synonym with the category "citizen at birth" by most people, including apparently Federal Judges.
The fact that the common use of language would indicate that these mean the same thing has no doubt helped this consensus to thrive.
Contra this de-facto and common understanding the birthers have always maintained that there are, in fact, three types of citzenship.
The birthers have for the most part not agreed on exactly what these extra requirements are. They include a mix of some of the following:
You would think that with all these very specific definitions the advocates would have some very specific laws to point to, but they don't. The term "Natural Born Citizen" is not defined in the US Constitution, nor is it elaborated on in US Code. So advocates were left with somewhat esoteric texts, such as 19th century French law dictionaries to make their case.
In the end it proved less than compelling.
Courts rejected all attempts to evaluate Obama using any of these additional criteria, so it appears that the de-facto definition is now the de-jure one as well.
But of course, birthers do not agree. And interestingly, Cruz fails every single additional test outlines above. So even though birthers don't agree on exactly what NBC means, they probably all do agree that Ted Cruz is not an NBC.
I suspect this will cause him as little problem as it caused Obama.
You are correct.
Ahnold was born an Austrian. Naturalized later. Ted Cruz was born an American. Thus, natural born.