Orthodoxy comprises several distinct groups, not just the two specified by Con.
The “Eastern Orthodox” are a group with similar beliefs, similar rules, and the same core liturgical praxis, united in faith, but not united in one earthly body. The Eastern Orthodox use two rites, Byzantine and Roman, with the Roman used only in a few parishes of the Russian and Antiochian churches. Several nominally Eastern Orthodox churches are not in communion with the rest; a few are in communion with some but not all of the others in their communion. Amongst them are several distinct subsets - Old Ritualists, the Ukrainian Kyiv Patriarchate - who are out of communion with the rest, but who are in almost all other ways identical.
The “Oriental Orthodox” are those churches in communion with the Pope of Alexandria. Most are Miaphysites - holding that Christ has two natures so tightly united that it’s hard to tell that they aren’t a third, separate nature.
The Old Believers are an odd mixture - those with priests and bishops are essentially Russian Orthodox, but rejecting the Nikonian reforms and those clergy who accepted them, tho some have come back into the Russian Orthodox Church as Old Ritualists. Other Old Believers are to Russian Orthodoxy as Protestants are to Catholics - Rejecting the system of Ordination, and often the sacraments, over ritualistic issues, then turning into a distinct theology.
The Independent Orthodox are vagante groups - many have only the vaguest ties to Either Catholic or other Orthodox churches. Most are tiny parishes with a bishop per parish, and often an auxiliary bishop or two, and a few priests, and a up to a few dozen parishioners. The Rest of Orthodoxy rejects them as non-orthodox. They directly parrallel the “Independent Catholics” in both the nature of parishes and their role to their parent churches. In the case of the Independent Orthodox, most such churches are of western-derived apostolic lineage, but Eastern Orthodox base praxis, often using Russian, Ruthenian, or Ukrainian liturgical texts.
The Assyrians broke away over the Nestorian controversy, but do not actually hold to Nestorianism. They are in limited pastoral communion with the Chaldean Catholics - that is, they permit the faithful of either to attend either for the common good.
The Jacobites - or Syrian Orthodox - are part of the Oriental Orthodox Communion, but it’s not always apparent.
The term Greek Catholic is usually used for churches in Union with Rome, but it’s also used in North America for some Russian Orthodox (OCA) parishes, and the OCA itself used to be the “Russian Orthodox Greek Catholic Church in America” (1924-1970), and in some out of the way parishes, the signage still hasn’t changed. Including some in Alaska.
The SSJK (Society of St. Joseph Kuntsephat) is an excommunicated group of Ukrainian Catholic clergy who rejected delatinization. They are affiliated with the SSPX, loosely, and use the term Greek Catholic as well.
Byzantine Catholic is used by both the Catholic Ruthenian Church and the vagante “Byzantine Catholic Church, Inc., Independent Jurisdiction” - the latter looks to be down to one or two parishes now, and at one point was at least 6.
“Latin Rite” has been used both to identify the Rite of Rome, and to identify the continued use of the 1962 missal, including in SSPX parishes, SSPV parishes, and some others. The SSPX are disobedient but still technically Catholic; the SSPV are heretics and excommunicated. It also technically applies to the Western Rite Orthodox use of the Russian and Antiochian Orthodox Churches in an academic sense, but they generally refer to that solely by “Western Rite” - and all such Orthodox parishes and/or liturgies are done under the auspices of Byzantine Rite hierarchs. Certain Western European Orthodox Churches were established by Russian Orthodox bishops, but are now “Independent Orthodox”, such as the French Orthodox Church, using the Latin-Rite “Western Rite” liturgy. Note also that Latin Rite has occasionally been used to describe Anglican liturgies done in latin according to the Sarum Missal.
One particular oddball group is the Celtic Orthodox Church - a couple parishes, using a restoration of the Celtic Rite liturgy from the Stowe missal, of Syrian Orthodox (Jacobite) Origin, but no longer in communion with them, and formerly in communion with the Romanian Orthodox. They are, however, in communion with some other Western Rite Orthodox Churches that are not in the Eastern Orthodox nor Oriental Orthodox Churches. Many of their clergy left and formed the British Orthodox Church, which is under the Omophor of the Coptic Patriarchate, and thus is once again in communion with the Syrian Orthodox…
There’s a reason “Orthodox” is often a dubious term… every christian church using the Nicene Creed claims to be both “One, Holy, Catholic and Orthodox” (perhaps not capitalized)… so many groups not in union with Rome, Constantinople, Moscow, nor Alexandria use one or the other, sometimes both, in their names.
