I don't wanna derail the informal roll call, but I have some questions regarding all these different groupings.
What is a:
Melkite Greek Catholic?
If answering these questions will derail the topic OR cause considerable mental anguish, please point me to the answer... preferably a website. Thanks!
A Latin or Western Catholic (pretty much synonymous terms) is one who belongs to what most people perceive as 'the Catholic Church' or 'Roman Catholic'. They are Catholics who worship according to the Latin Rite (or one of its few variants). Prior to VII, they would have worshipped at Mass in Latin; subsequent to VII, most would have attended Mass served in the vernacular of their country; more recently, they might do either -depending on their locale and the availability of Mass served in Latin.
Montanist is a term not properly used in this context, at least historically. It is, I suppose, intended to define one as being supportive of the prerogatives of the papacy, but not carried to what some would consider extreme (yet would be viewed by others - Ultramontanists - as its proper and fullest expression). (The term, in its original usage, referred to the Montanist heresy, which was not about papal prerogatives.) Those who use it as I described above would be doing so to distinguish themselves from Ultramontanists - but to understand what it meant to any who so called himself, you'd have to inquire further, as it hasn't got as defined an understanding as Ultramontanist does.
Ultramontanists are generally thought of as those Catholics who zealously support, defend, and promote papal authority and supremacy over conciliarism. (One could be an Eastern or Oriental Catholic - discussed below - and be an Ultramontanist, but they are more generally found among Latin Catholics.) I would agree with Papist that an Ultramontanist view of the papacy is a negative one, but I admit to being surprised that he perceives it as such - that would definitely be the Orthodox viewpoint and one found among at least some Eastern and Oriental Catholics (I'd say most, but I have no idea how one would quantify that, so I'll refrain from doing so).
Eastern Catholics is the collective term for those Catholics, in communion with Rome, who worship according to the Byzantine or Constantinoplian Rite. They belong - in all but one instance - to Churches sui iuris
(of their own law) to which there is an Eastern Orthodox counterpart, as they entered communion with Rome from Eastern Orthodoxy at some point subsequent to the events of 1054 and thereafter. There is a stickied thread here
which delineates the 14 Eastern Catholic Churches (although it may be out-of-date on a few points, since I wrote it about 3 years back).
Not referenced above are Oriental Catholics. That is a collective term for those who belong to the any of the 8 Catholic Churches which have counterparts among the Oriental Orthodox Churches. They are also listed in that same sticky.
There are also 2 other non-Latin type Catholics - Maronite and Chaldean - which fit into neither the Eastern nor Oriental classification - the Maronites because there is no counterpart Church in Orthodoxy and the Chaldeans because they represent those who entered comunion with Rome from the Assyrian Church, which belongs to neither of the Orthodox Communions.
Melkite Greek-Catholics are members of the Church which was formed after splitting from the Greek Orthodox Patriarchate of Antioch (whose faithful were then known as Melkite Orthodox and are now known as Antiochian Orthodox) back in 1724-25. They are one of the 14 Churches collectively termed as Eastern Catholic.