Eastern Orthodox Catholic. It's an important distinction. Roman Catholicism and Eastern Orthodox Catholicism regard each other as interchangeable and have for a long time. You can recieve communion at either, go to confession at either, and the other will regard it as perfectly valid. That's why they were lumped together as simply "Catholicism". Anglicanism has also been retroactively lumped into the same category by the Pope's olive branch to the Anglican community; they're being allowed to keep their own practices if they choose to change their allegiance which amounts to tacit approval of their practice, and the Patriarch or anyone else has not issued any modification of the approval of Roman Catholics to exclude the Anglicans who change allegiance but not practice so that is tacitly approved as well. The Lutherans fall into a somewhat grey area in that Eastern Orthodox Catholic and Roman Catholic will allow communion but not confession. That pretty much accounts for 70%+ of Christianity right there.
So as a practical matter, the majority of Christianity sees it in a much less complex light then you do. Two of what you would term "major factions" literally consider themselves brothers.
"Catholic" is an attribute claimed by any communion that believes itself to be representative of the ancient, undivided Christian church, and thus necessarily puts those organizations that use it at least somewhat at odds with one another, though they certainly do sometimes make good-faith efforts to broaden their common ground. The smooth interchangeability you refer to is between the Roman Catholic Church and the autonomous Eastern Catholic Churches, which are in full communion with the Pope but often use Orthodox rites. These churches are distinct from what we commonly call the Eastern Orthodox Church, which refers to itself as the Orthodox Catholic Church, and is most certainly
not in full communion with the Pope. Attempts at reconciliation between Roman Catholicism and the Orthodox Church are relatively recent; if I'm not mistaken, John Paul II was the first bishop of Rome to visit Eastern Orthodox territories since the East-West Schism, and he was not always greeted warmly on his tour.
But I take your point that most Christians themselves probably do not see it as quite so complex. And as I was originally trying to get at, the question at the center of this discussion is not religion's tendency to fracture in abstract space, but rather its tendency to engage with local and regional cultures, resulting in fractures that do not necessarily coincide with official divisions. My original example of a discussion between an Ethiopian Orthodox priest and a Pentecostal minister was not meant to highlight the differences between catholic Christianity and Non-Trinitarianism, (vast though they may be), but to draw an analogy to Islam's regional heterogeneity. What I mean to say is, the greatest differences between Christians you will find are often geographically based. (And these geographical differences inform, and are informed by, denominational distinctions.) Islam in Indonesia is substantively different from Islam in Morocco, and Christianity in Ethiopia is substantively different from Christianity in the United States. That's all I was trying to say.
