Example: Do the Orthodox call themselves Eastern Orthodox or Orthodox?
Because the Orthodox consider their communion to be the One, Holy, Catholic, and Apostolic Church, their church’s
official name is actually the
Orthodox Catholic Church. Another common name they use for their faith is “Holy Orthodoxy.”
Because of their church’s actual name, some will go around calling themselves “Orthodox Catholics,” although for the sake of eliminating confusion (“Wait - you’re Catholic? What?”) most will simply call themselves Orthodox.
Those Orthodox who do prefer to call themselves Orthodox Catholics will refer to Catholics like you and me as “Papal Catholics” or “Roman Catholics” or just “Latins.”
If it’s Eastern Orthodox, then what happened to the Western Orthodox?
Interestingly enough, there are western-rite Orthodox in existence: some Orthodox communities in the western world have devised a rite based on the Tridentine Mass, but with an explicit epiclesis.
These “western-rite Orthodox” are
not analogous to the eastern Catholic churches, though, for two reasons: (a) they’re not hierarchically distinct
at all, and (b) they weren’t formed out of some already existing community but were basically just invented.
The earliest names given to those who believed in Jesus of Nazareth was…“Church of God”…“Church of Christ”…“Bride of Christ”…“Body of Christ”…“Church of the Firstborn”…“Temple made with Living Stones”…In all seriousness is anyone concluding that because an early church father used “universal ecclesia”…“catholic church”…THAT became the formal name AND because the Catholic church uses the “Catholic” part in their name…that this somehow proves the Catholic church of today is what the ECF had in mind when he wrote “catholic church”?
No, we’re not claiming that.

It is, however, true that the Catholic Church can indeed trace itself seamlessly back to the Apostles. We are not the only communion that can do this, though - the Orthodox, Oriental Orthodox, Assyrian Church of the East, and a couple others can as well.
So the Catholic Church’s position that she is the very same church that Saint Ignatius of Antioch spoke of as “the Catholic Church” is not at all a far-fetched claim.
I know but Mathew Holford implied Roman was incorrect when he said Latin is correct instead.
“Roman” isn’t necessarily wrong, but “Latin” is better because it’s more specific. Here’s why:
If you speak of the
Latin Church, it’s obvious what you’re referring to.
If you speak of the
Roman Church… is that the whole
Latin Church or do you actually mean the
Diocese of Rome?
Frankly, whenever non-Catholics of a more polemical bent - whether Orthodox or Protestant - speak scathingly of “the Church of Rome,” I just roll my eyes and inform them that I’m a member rather of the Church [Diocese] of Columbus, OH. The bishop of my particular church, however, is
in communion with the bishop of Rome - which makes my church part of the whole Catholic Church as well.