Just found out what Adrian Fortescue said in The Mass: A Study of the Roman Liturgy (1914):
Mass (missa) has become the proper name for the Latin liturgy. Its first certain occurrence is in a letter of St. Ambrose, where it is the liturgy of the faithful only. [Footnote: Ep. i, 20, 4-5 (P.L. xvi, 995).] But it is not for some time used exclusively for the Holy Eucharist. Its meaning and derivation, once much discussed, are not really doubtful. It is a late Latin form for missio [So Collecta, Ascensa, Ingressa, Confessa, etc.] and meant originally merely “dismissal”. Avitus of Vienne († 523) uses it for the dismissal from churches or law-courts in the most general sense: “missa fieri pronuniantur” (= the people are dismissed). [Ep. I quoted by Rottmanner: Ueber neuere u. ältere Deutungen des Wortes missa; Tübinger Qtlschr. 1889, pp. 532 seq.] So it occurs constantly for the dismissal of the catechumens in the Eucharistic service. St. Augustine, for instance: “post sermonem fit missa catechumenorum.” [Sermo lxix, 8 (P.L. xxxviii, 324).] A Synod at Lerida in Spain (524) says that people guilty of incest may remain “usque ad missam catechumenorum,” namely till the catechumens are dismissed. [Can. 4. Hefele-Leclercq: *Hist. des Conciles ii, 1064.] St. Benedict († 543) in his rule uses *missa for the dismissal from the divine office too. [Cap. xvii.] As there was a dismissal of the catechumens, so after Communion there was a dismissal of the faithful (“Ite missa est”). Florus of Lyons († 860) explains the word exactly: “Missa nihil aliud quam dimissio, id est absolutio, quam celebratis omnibus tunc diaconus esst pronuntiat quum populus a solemni observatione dimittitur … Tunc enim, clamante diacono, idem catechumeni mittebantur, id est dimittebantur foras. Missa ergo catechumenorum fiebat ante actionem sacramenti; missa fidelium post confectionem et participationem.” [size=]de actione missae, n. 92 (P.L. cxix, 72)][/size] From this a transition to meaning the whole of each part of the service was easy. To stay till the missa catechumenorum or fidelium became to stay for the missa. We have then many texts which speak of these two missae as the two parts of the liturgy. [E. gr. Ivo of Chartres († 1116) Ep. 219 (P.L. clxii, 224)] The Peregrinatio Silviae constantly uses “missa” for the liturgy of the faithful. [E. gr. xxiv, 11 etc.] Innocent I (401-417) [Ep. xvii, 5 (P.L. xx, 535)] Leo I (440-461) [Ep. ix, 2 (P.L. liv, 627)] in the same way. The disappearance of the discipline of the Catechumenate made a distinction between these two missae meaningless, so we find then the word used simply for the whole function. The Leonine Sacramentary supposes the word throughout; “Item alia” means “alia missa”; and the Gelasian book uses it constantly. [E gr. “Orationes et preces ad missam” (ed. Wilson, p. 29), “missa chrismatis” (p. 69), etc.] But a plural form, “missae,” “missarum solemnia” (for one Mass) remains in the middle ages, perhaps as a memory of the old two “masses,” of the catechumens and of the faithful.
It is not really surprising that so, step by step, the name of an unessential detail should have become that of the whole service. Liturgical language offers many similar examples. [For instance our common use of “Breviary” for the office, “Maundy Thursday” etc. Even “Confession” is not really the most essential element of the Sacrament of Penance, and so on.] The points to remember about the word Mass are, first, that it is not an essential name for the Eucharistic sacrifice, used everywhere from the beginning. It is a late term arising almost by accident in the West only. Except for later associations “Mass” no more involves the idea of sacrifice than do such names as “Lord’s Supper” or “Communion Service.” Secondly, we should never use the word for an Eastern rite. In the East they have the older technical term “Liturgy,” certainly at least equally significant. Mass is not a general name used everywhere and connoting a theological idea. It is the name this function acquired in the Roman and Gallican rites only.
So yeah, basically what Ad Orientem said earlier.