What's very clear is that this question/subject is as divisive as it is difficult to understand. If you look up American Nationality on Wikipedia the answer is as unclear as it is in this forum. I can tell you this; my grandparents were born in Turkey and Armenia. I've always referred to myself as an Armenian. I can understand though why those without a distinct cultural or ethnic background would call themselves simply Americans. To be technical, I can certainly understand why I could say I'm an Armenian American. It's not a conscious thought I have when asked what I am. I suppose it depends too how you're raised. With me, I attended a private Armenian school from K-10 as well as an Armenian church. Basically, everyone I knew growing up was Armenian. It was kind of like My Big Fat Greek Wedding, except Armenian.
Simply put, I think it's just a matter of symantics. My race is caucasion, I'm a citizen of the United States of America, and my bloodline is Armenian. Does it really matter how I choose to describe myself to someone else?