The two do not necessarily exhaust the possibilities. I agree that they were the Church. Also it seems that your side has trouble taking seriously our Lord's identification with his followers, with the Church which is His body.
You say Rome "claimed" the Church. We say it alighted there. Your side often offers this idea of an imperialist papacy. Fortunately I do not find repetition an effective tool of persuasion.
Christ’s church rode off in all directions. If any of it “alighted” at Rome, so did it alight and catch fire elsewhere. In fact Alexander the Greek did more for it than Rome ever did.