It is proper to refer to the Eastern Catholic Churches, in the plural. Each of the Eastern Catholic Churches are a full church in their own right (not RITE>>>>LOL). The Catholic Church is a communion of CHURCHES.
You are correct. Some Churches share the same rite and some Churches have multiple rites.
Alexandrian liturgical tradition; 2 sui iuris Churches
Coptic Catholic Church
Ethiopic Catholic Church
Antiochian (Antiochene or West-Syrian) liturgical tradition; 3 sui iuris Churches
Maronite Catholic Church
(West) Syrian Catholic Church
Syro-Malankara Catholic Church
Armenian Rite; 1 sui iuris Church
Armenian Catholic Church
Chaldean or East Syrian liturgical tradition; 2 sui iuris Churches
Chaldean Catholic Church
Syro-Malabar Catholic Church
Byzantine (Constantinopolitan) liturgical tradition; 14 sui iuris Churches
Albanian Greek-Catholic Church
Belarusian Greek Catholic Church
Bulgarian Greek Catholic Church
Byzantine Church of Croatia, Serbia and Montenegro
Greek Byzantine Catholic Church
Melkite Greek Catholic Church
Hungarian Greek Catholic Church
Italo-Greek Catholic Church
Macedonian Greek Catholic Church
Romanian Church United with Rome, Greek-Catholic
Russian Catholic Church
Ruthenian Catholic Church
Slovak Greek Catholic Church
Ukrainian Greek Catholic Church
The Latin (Western) Catholic Church 1 sui iuris Church that celebrates many liturgical rites
Actively celebrated:
Roman Rite, whose historical forms are usually classified as follows
Pre-Tridentine Mass (the various pre-1570 forms)
Tridentine Mass (1570-1970 and still authorized in circumstances indicated in the document Summorum Pontificum as an extraordinary form of the Roman Rite)
Mass of Paul VI (1970–present)
Anglican Use (restricted to formerly Anglican congregations)
Ambrosian Rite (Milan, Italy and neighbouring areas)
Aquileian Rite (defunct: northeastern Italy)
Rite of Braga (Braga, Portugal)
Mozarabic Rite (Toledo and Salamanca, Spain)
Defunct or rarely celebrated:
Durham Rite (defunct: Durham, England)
Gallican Rite (defunct: Gaul, i.e., France)
Celtic Rite (defunct: British Isles)
Sarum Rite (defunct: England)
Catholic Order Rites (generally defunct)
Benedictine Rite
Carmelite Rite
Carthusian Rite
Cistercian Rite
Dominican Rite
Franciscan Rite
Friars Minor Capuchin Rite
Premonstratensian Rite
Servite Rite