Judaism American Jew or Jewish American

[Jason]

Actually, my question had nothing to do with citizenship. Nor did my question have anything to do with differentiating between citizens. This was not about hyphenating Americans. 

The question I posed was this, "Are you an American Jew or a Jewish American?" Keep in mind that the question was posed in the context of the class subject. If you don't understand the question, I will explain.

Yes, please explain.  Thank you.

...Or, don't.

LOL

Essentially, the question is about how you identify primarily. By calling yourself an "American Jew," you identify first as a type of Jew (specifically, an American one). By identifying as a "Jewish American," you identify first as a type of American (a Jewish one). It's a question of essential identity.

In Israel we talk mostly about "American Jews" rather than "Jewish Americans," as far as I'm aware. That is, at least in the sources that I listen to. יהודים מאמריקה "Jews from America." That would be opposed to something like אמריקאים עם רקע יהודי "Americans with a Jewish background."