On one level, I don't see the point of this. What's wrong with being Near Eastern and Middle Eastern?
For the purpose of the discussion, however, for much of history the west was indeed defined as "Christendom", and whether people are aware of it or not, or like it very much, Christianity has molded the world view of the west. Armenian and Syrian and other "eastern" Christians share part of that world view. One can look at the period of the Crusades for examples of how this worked in "real life". The moronic Crusaders at first slaughtered the eastern Christians because they didn't realize they were co-religionists. Any "native" wearing "eastern" clothing was killed. They shortly realized differently, however, and there was cooperation and intermarriage, notably with Armenians. On the other hand, the Armenians as a people did not participate in the whole long chain of development of western culture, and so they straddle the line a bit.
Fwiw, the Armenians in my area are not considered Middle Easterners. I think it was different in California, so it's a bit ambiguous. Also, Syrian Christians who have been settled here through the work of various aid organizations including Catholic Charities adjust much more quickly and assimilate more quickly than Muslims by far, unless the Muslims are secularized, as is the case with some Turks.
The only way to determine how close Armenians are genetically to Europeans is to do something like an fst analysis. Anything else involves a whole raft of subjective assumptions.
For example, someone could compare the fst scores of Armenians to Germans versus Central Asians like Pashtuns to Germans.