I think what you answered makes sense. It sounds like you have a strong background that transcends the "I'm part Irish/German/Italian/This/That/And-The-Other-Thing" that a lot of Americans have. I think my family is so far removed from whatever our origins are that there's really nothing else to put except "American". Plus, being of Jewish descent is considered as being part of a specific ethnic group genetically rather than just a religion. It's kind of really interesting to think about. 
Same for me. I grew up hearing Yiddish spoken at home.
I don't want to put Jewish as my religion because I have issues with organised religion in general, and I don't want to be lumped in with Jews who think that married women should keep their heads covered, that women are unclean when they are menstruating, and that God will smite me if I eat a pork pie.
I also don't want it to be assumed that I support government funding for teaching these beliefs in schools.
I do, however, want my ethnic and cultural identity to be recognised.
ETA: When I lived in the US, it always seemed to be politically incorrect to call yourself American when you mean you were from the United States, because you were ignoring Canada and Latin America. (Manhattan has an Avenue of the Americas - in honour of all the American nations.)
If someone asked me what country I was from, I would say the United States; I would never say America or that I was American because I thought that would be offensive.
I didn't start referring to myself as American until I moved to the UK and other people started calling me American.
About race/ethnicity - these distinctions are purely cultural/traditional and have no scientific basis and will change from country to country.
For example, in the US I always considered people from India, Pakistan or Bangladesh to be white, and in the UK they are considerd Asian.
And the concept of Hispanic (as a race/ethnic group that is distinct from black or white) doesn't exist in the UK.