Rites refers to the distinctive customs of each particular church, both individually (I.e. The Armenian Baptismal Rite) and as a whole (the Armenian Rite); whereas one could also refer to the Rites of the Christian Church in comparison with the Rites of Islam, the Rites of the Mandaeans, et cetera. Rite and ritual are somewhat interchangeable but in liturgiological jargon Rite tends to refer primarily to national or regional ritual usages: the East Syriac Rite, the West Syriac Rite, the Ambrosian Rite, the Roman Rite, the Mozarabic Rite, the Byzantine Rite, the Jerusalem Rite, now defunct, but the source for much of our current liturgical heritage including nearly everyone's Paschal services, which can be traced back to St. Cyril of Jerusalem, to whom fell the task of organizing the Pascal pilgrimmages, devotions and liturgical services following the restoration of the city under St. Helen (he had one or two predecessors but a large chunk of the ritual started with him if memory serves; recall prior to Ss. Constantine and Helen Jerusalem was an administrative center off limits to Jews and Christians and mostly in ruins). Likewise the Antiochene Rite, which had wide ranging influence on the Byzantine Rite and West Syriac Rites, and all the Rites they influenced, including the Armenian and Ethiopian; traces of this ancient Rite which is perhaps the oldest can even be found in the Roman Rite. However this rite and the rite of Jerusalem were largely lost in their original Greek forms due to Byzantinization and are best preserved in the Armenian and Syriac Orthodox liturgies. Of course the Byzantine Rite is itself derivative of the Antiochene Rite and the Jerusalem Rite, with some local influence and a theoretical Cappadocian influence.
Lectionary refers to the system of appointed scripture lessons throughout the year. In the Armenian liturgy these are fairly simple due to the relative lack of fixed feasts, but the intersection of fixed and movable feasts, in particular the Lent-Pascha-Pentecost block I like to call the Triodia makes this incredibly complicated in the Byzantine and pre-1910 Tridentine Rite. Before the reforms of Pius X the calendar of saints had grown to the point that outside of privileged octaves and seasons such as Advent, Lent and Holy Week, the proper Sunday lection was invariably never heard, displaced by various saints days.
Now by the way I will say I love the Armenian liturgy. However thanks to the wicked Turks who perpetrated genocide against all the Christian populations of the Ottoman Empire starting in 1915, much of this has been lost. I refer of course to the older style of Armenian liturgical music which you can still hear in the monastic chanting of the psalms but not in the music of the Sunday Badarak, which has been heavily westernized. In the recent past this was accompanied by cymbals in the manner of the Syriac Rites (for some reason the Chaldeans and the Syriac Catholics kept the cymbals while jettisoning nearly everything else whereas the non-Roman churches lost them, although given the, to my knowledge, lack of cymbals in the Indian church it's possible the reports I've read from the 19th century of cymbals in use in Syriac Orthodox parishes were erroneous or referring to a regional or tribal custom that was prevalent in that part which left for Rome but was otherwise destroyed by the Turks.
Some object to the Byzantinization of the Liturgy of the Catechumens and the Latinization of the mitres and the Last Gospel, but I don't.
However there were once 13 or so Anaphoras that we have the Classical Armenian text for in use but only the Anaphora of St. Athanasius remains in use. Now granted, the Anaphora of St. Athanasius is, in American vernacular parlance, a hum-dinger; Athanasius is my co-patron, my avatar is a Syriac Orthodox icon of him, and I love the text of the Armenian Anaphora of St. Athanaius so much that I might well inscribe it on my wall (actually although I can't read or write Classical Armenian the text of the Badarak inscribed in faux-granite would make an awesome decoration for any Armeniophile). But I would really love to see the Anaphora of St. Ignatius,moor example, which is believed to be an original Armenian composition, restored to use in Armenian churches.
By the way, I do love the sound of Westernized Armenian music. But I wish the more exotic sound with cymbals and so on were also available. Imagine the liturgical thrill of using them side by side! I also adore the look of old Armenian churches in the East ranging from the Cathedral in Isfahan to the heavily Armenian influenced Churches of the Nativity and Holy Sepulchre. Imagine the heart pounding thrill of attending such a vibrant, living Armenian Badarak, in a church dimly lit through stained glass windows and with a vast array of hanging silver and brass oil lamps, with huge clouds of incense obscuring the vision, so that the icons on the bema, illuminated with candles, penetrate through a holy fog, to hear a mixture of the newer music with the organ, and the old, with the clash of cymbals, perhaps in a mix of Classical Armenian and the vernacular, with the 13 ancient anaphoras rotated through each week. Such a church would do more than attract Armenians, I'll wager it could single handedly wipe out a heretical church like St. Gregory of Nyassa Episcopal Church in San Francisco and rescue hundreds of Americans from heresy.
This is the fantastic realm of liturgical studies: to spend ones Christian devotion with the mind uplifted perpetually in the space between Heaven and Earth, by continually researching and contemplating the liturgical life of the entire church past and present. Mand whereas many liturgiologists take the approach of wanting to simplify or innovate, redoing the liturgy to suit their own personal convictions, I swear that until the last breath I will do all that I can ethically do in the service of our Lord to help the Orthodox and the semi-Orthodox (I.e. Roman Catholics, Assyrians) from losing their liturgical heritage or having it disfigured by modernists, and also to provide thoughts on how those parts that already have been lost, like the disused Armenian anaphoras, can be restored.