As a side note for our Armenians friends...
What would your reaction be if someone refused to call you Armenian and henceforth you are Hayastani. And it is the Hayastani Apostolic Church in English. Imagine the reaction it would cause!
I think you mean Hayastantsi. (The "tsi" ending often is used to denote nationality. Like Germans are "Kermanatsi.")
My reaction would be to drop dead in shock that a non-Armenian was trying to speak Armenian. I'm very impressed by you, by the way.
The Armenian word for an ethnic Armenian person is Hay (pronounced like "high.") It means the exact same thing as "Armenian," it is just two different languages. Hay and Armenian are both good. Hayastantsi is usually used specifically for someone from the Republic of Armenia. Hay is more general.
In Armenian, the Armenian Church is often called "Hay Yegeghetsvo." If someone mixed Armenian and English and called our church the "Hay Church," it would be fine. Someone would just assume you were an American Armenian who mixed his Armenian with English, like I often do. Sometimes my friends and I get very bad about it. Like if you want to hang up your coat in a closet, you'll say "coat-us closet-een mech bidi hang-up-enem." Once I recall a friend of mine, in reference to Palm Sunday, saying, "Hisoos-uh donkey-een vrah ride-erav." Of course, we could speak it more properly if we wanted to really try, but it is so much easier to take English words and add the Armenian endings to them.
Where was I? Oh yeah. I don't think you'll find an Armenian equivalent to the Greek desire to be called Roman. It's probably why I have trouble understanding it. We're happy to just be called Armenian (or Hay.)