It seems that by the 800s, the Catholic Church had to start making proclamations in vernacular languages even in places where Romance languages were spoken, as the languages had diverged so far from Latin and even from Vulgar Latin.
But what about the city of Rome itself? Did the language spoken by the common people and even by the Church's officials stay as "Latin", or did it have a different name by the year 1000 (like maybe "Roman", analogous to "Tuscan" or "Venetian")?
They probably would have called what they spoke "Latin" or the "Roman tongue", but there was a slowly-growing awareness from ca. 950 on in the Italian peninsula that what they were speaking was not, in fact, Latin, at least among intellectuals. So, by 1000, you are almost certainly beginning to see a differentiation, although Latin remained intelligible to Italian speakers until the 13th c.
See:
The liturgical Latin as we know it was based on 'golden' age Latin, developed during the middle Carolingian period. Even during the imperial period, the 'Latin' spoken in the western part of the Empire would have varied wildly, although it seems likely they would have been relatively mutually intelligible. Indeed, the Latin as written by Golden Age authors would have not had much to do with the Latin spoken even by the plebs in Rome, let alone the people of Gaul or Hispania. George Holmes called the choice by Carolingian church leadership to embrace and codify that particular branch of Latin as misguided even at the time at any rate. As telkanuru mentioned, the 'Roman tongue' is generally what the Germanic leadership would have called the various forms of vulgar Latin that existed in their Empire, and the people's of Italy would still had considered as such.