There is the Eastern Orthodox Church (Russian Orthodox, Greek Orthodox, Bulgarian Orthodox, Georgian Orthodox, Ukrainian Orthodox, etc.) - This is all one Eastern church.
There is the Oriental Orthodox Church (Coptic Orthodox, Syriac Orthodox, Ethiopian Tewahedo Orthodox, Malankara Orthodox, etc.). This is all another Eastern Church.
There is also the Assyrian Church of the East, but ever since the war in Iraq and rise of ISIS, it's all but destroyed at this point.
Between those 3, those are the only "eastern" churches in existence apart from tiny schismatic groups. However, when someone refers to the Eastern Church, they are almost always referring to the first one, the Eastern Orthodox Church, since it's enormous and sat at the center of the Byzantine Empire.
In contrast, there is single Catholic Church, and just a century prior to Gregory's time it united all Christians in the Western Europe, apart from a few heretical / protestant movements here and there.
at the height of the Church’s calendar problem, in the second half of the 16th century, the eastern Church and the western Church were an incredible ten days out of sync with one another. This was only reconciled in 1582 when Pope Gregory XIII implemented what has become known as the Gregorian calendar reform.
This doesn't sound right at all. Why would the church calendars be out of sync at the time? In fact, it was the Gregorian reform that put them out of sync. They are still out of sync and the gap is growing.
Bonus history trivia question: What month did the Russian October Revolution happen?