When a group leaves the Church, they end up with a new name. For example, the up-starts in Germany became Lutheran. The Swiss became Calvinists. Then, when those groups split, the new one takes a new name as well.
Now, if the Catholic Church left Orthodoxy, why did we keep the name? Why did we not become something else and they retain the name CAtholic?
In most cases, the names churches end up with were put on them by their detractors, this is true of Lutherans, Calvinists, Methodists and a lot of others.
In all these cases, while the names have stuck, the adherents themselves preferred other names.
But even this business of nomenclature is not a Protestant innovation.
The name Catholic, though very old, is not original, it appears to have developed during the Christological controversy to designate those believers which held to the Nicene definition and was itself a taxonomic choice made to define a specific theological position.
The name Catholic Church has been around since even before the time of Saint Augustine. It was an old name even by than. So the official name of the Church did not start in the Great Schism as some like to think.
Also, the dispute between the Orthodox and Catholic Church is rather interesting, name-wise, because some people think that there were no conflicts in the Church before the Great Schism. Yet it is evident from the Epistle to Timothy that the Church was plagued with false teachers and heresies ever since her humble beginnings. So I think the Church’s official name came early on to distinguish her from other churches and to show what she was: the Catholic Church.
Because neither Catholicism nor Orthodoxy exactly “left” the other.
Of course, the names you list were generally given by opponents originally. Whether a group manages to stick with a more complimentary name (Catholic, Orthodox, Evangelical, Reformed, Christian, Apostolic, Pentecostal), or is stuck with a name deriving from a human founder (Lutheran, Calvinist, Wesleyan), is really a matter of historical circumstance and not of huge theological importance–though the rather rare circumstances where Christian traditions actually embrace names deriving from a human founder are significant and somewhat disturbing (I see this among Wesleyans to some extent).
Well, when I was in communion with Rome, I was called by a variety of titles. Latin Catholic, Roman Catholic, Catholic. When I was in the Eastern Catholic Church I was called: Byzantine Catholic, Ruthenian Catholic, Eastern Catholic, Roman Catholic, Catholic.
And now to back up my claims about how old the name “Catholic Church” is. Here are the Church Fathers on the Church:
Wherever the bishop shall appear, there let
the multitude [of the people] also be; even
as, wherever Jesus Christ is, there is the
St. Ignatius, Epistle to the Smyrnaeans
We say that both in substance and in
seeming, both in origin and in development,
the primitive and Catholic Church is the only
one, agreeing as it does in the unity of one faith
Clement of Alexandria
And if ever thou art sojourning in any
city, inquire not simply where the Lord’s
house is–for the sects of the profane also
attempt to call their own dens, houses
of the Lord–nor merely where the church
is, but where is the Catholic Church.
For this is the peculiar name of the holy
body the mother of us all.
St. Cyril of Jerusalem, *Catechetical Discourses *
Now it [the Church] is called Catholic
because it is throughout the world, from
one end of the earth to the other.
St. Cyril of Jerusalem
Whether they wish or no, heretics have
to call the Catholic Church Catholic