Patriarch Ignatius Joseph III Yonan of the Syriac Catholic Church, an Eastern Catholic church based in Damascus, Syria, as seen during a conference about the "extermination of Christians in the Near East" (2015).
The Eastern Catholic churches are instead distinct particular churches sui iuris, although they maintain full and equal, mutual sacramental exchange with members of the Latin Church. Apart from its reference to the liturgical patrimony of a particular church, the word has been and is still sometimes, even if rarely, officially used of the particular church itself.
The Church of the East, which was mainly under the Sassanid Empire, never accepted the council's views.
The Second Vatican Council spoke of Eastern Catholic Churches as "particular Churches or rites".
In 1999, the United States National Conference of Catholic Bishops stated: "We have been accustomed to speaking of the Latin (Roman or Western) Rite or the Eastern Rites to designate these different Churches.
"Monophysite" implies a single divine nature alone with no real human nature - a heretical belief according to Chalcedonian Christianity - whereas "Miaphysite" can be understood to mean one nature as God, existing in the person of Jesus who is both human and divine - an idea more easily reconciled to Chalcedonian doctrine.
They are often called, in English, Oriental Orthodox Churches, to distinguish them from the Eastern Orthodox Churches.
Thus the term Latin rite can refer either to the Latin Church or to one or more of the Western liturgical rites, which include the majority Roman Rite but also the Ambrosian Rite, the Mozarabic Rite, and others.
In the 1990 Code of Canons of the Eastern Churches (CCEO), A group of Christian faithful linked in accordance with the law by a hierarchy and expressly or tacitly recognized by the supreme authority of the Church as autonomous is in this Code called an autonomous Church (canon 27)." (canons 111 and 112), and also speaks of "a subject of an Eastern rite" (canon 1015 §2), "Ordinaries of another rite" (canon 450 §1), "the faithful of a specific rite" (canon 476), etc.
They are most properly called Eastern Churches, or Eastern Catholic Churches." forbids a Latin bishop to ordain, without permission of the Holy See, a subject of his who is "of an Eastern rite" (not "who uses an Eastern rite", the faculty for which is sometimes granted to Latin clergy).
The term Uniat or Uniate applies to Eastern Catholic churches previously part of Eastern or Oriental Orthodox churches or of the Assyrian Church of the East.
The Churches that refused to accept the Council considered instead that it was they who were orthodox; they reject the description Monophysite (meaning only-nature) preferring instead Miaphysite (meaning one-nature).
The difference in terms may appear subtle, but it is theologically very important.
However, the Church's contemporary legislation as contained in the Code of Canon Law and the Code of Canons of the Eastern Churches makes it clear that we ought to speak, not of rites, but of Churches.