From the Catholic Encyclopedia:
In the East, when a Church is spoken of, four things must be kept distinct: the race to which the adherents of the Church belong; the speech used in their everyday life, and in their public devotions; the ecclesiastical rite used in their liturgy, and their actual belief, Catholic or non-Catholic. It is because these distinctions have not been, and are not, even now, always observed that a great confusion has arisen in the terminology of those who write or speak of the Eastern (Oriental) Churches and of the Greek Church. As a matter of fact, the usual signification attached to the words Eastern Churches extends to all those Churches with a liturgical rite differing from the Latin Rite. Let them reject the authority of the pope or accept it, they are none the less Eastern Churches. Thus the Russian Church, separated from Rome, is an Eastern Church; in the same way the Greek Catholics who live in Italy, and are known as Italo-Greeks, make up an Eastern Church also.
The expression Eastern Churches is therefore the most comprehensive in use; it includes all believers who follow any of the
six Eastern rites now in use: the Byzantine, Armenian, Syrian, Chaldean, Maronite, and Coptic.
What, then, do we mean when we speak of the Greek Church? – Ordinarily we take it to mean all those Churches that use the Byzantine Rite, whether they are separated from Rome or in communion with the pope, whether they are by race and speech Greek or Slavs, Rumanians, Georgians, etc. The term Greek Church is, therefore, peculiarly inappropriate, though most commonly employed. For instance, if we mean to designate the rite, the term Greek Church is inaccurate, since
there is really no Greek Rite properly so called, but only the Byzantine Rite. If, on the other hand, we wish to designate the nationality of the believers in the Churches following the Byzantine Rite, we find that out of fifteen or twenty Churches which use that rite, only three have any claim to be known as The Greek Church, viz., the Church of the Hellenic Kingdom, the Church of Constantinople, the Church of Cyprus. Again, it must be borne in mind that in the Church of Constantinople there are included a number of Slavs, Rumanians, and Albanians who rightly refuse to be known as Greeks…
The Greek Orthodox Churches are Churches separated from Rome and following
the Byzantine Rite, i.e. the rite
developed at Constantinople between the fourth and tenth centuries. In the beginning, the only language of this rite was Greek.…
There are five divisions of the Byzantine Rite, and consequently five divisions of Orthodox Greek Churches:
- The Greek-Byzantine Rite, which includes the pure Greeks subject
o to the Patriarchate of Constantinople,
o to the Holy Synod of Athens, and
o to the Archbishopric of Cyprus.
- The Arabic-Byzantine Rite, which includes the Christians under the Patriarchates of
o Antioch,
o Jerusalem,
o Alexandria, and
o the Archbishopric of Sinai.
- The Georgian-Byzantine Rite, which, up to the beginning of the nineteenth century, included the Churches of the Caucasus Range now absorbed by the Russian Church and obliged to use the Slavonic Liturgy instead of their own native Georgian.
- The Slavonic-Byzantine Rite, comprising
o the Russian,
o the Servian, and
o the Bulgarian Churches.
- The Rumanian-Byzantine Rite, used by the Rumanian Churches.