Similarly, when did the inhabitants of other Romance language nations such as Spain and Portugal stop referring to the vernacular language as Latin and start referring to it by another name?
Already in Roman times, each part of the empire had begun to develop a different, "Vulgar" form of Latin that would go on to develop into the various romance languages. However, I'm not a historian of late Roman history (although I find the time period fascinating) nor am I a linguist. However, I do know a fair bit about the development of the Italian language, especially during the renaissance.
As late as the time of Dante Alighieri (the early 14th century), the languages commonly spoken by the people in Italy were still being called Lingua Volgare, or "Vulgar Language", and was seen as subordinate to Latin. I wrote an answer to a similar post on he subsequent development of the Italian language here, perhaps it can answer some of your questions.
It is worth noting that for most of Italy's pre-unification history, the language called "Italian" was a much-modified form of the Vulgate spoken in Florence, and existed only as a literary language, used only by a certain clique of elites. The inhabitants of the various regions would have spoken and written in the various regional dialects (which oftentimes have individual variants within the same region). Only with the advent of Radio, and Television after it, would the Italian language begin to be widely spoken.
Through Roman colonization, Latin was hugely influential on the Iberian Peninsula as well. Wright distinguishes between Late Latin as Latin in written form, and as an early Romance in oral form. Initially, the Spanish Romance vernacular was also known as ladino, which came to be replaced during the 12th century by terms like romance, castellano etc. The Poema de Mio Cid is commonly seen as the starting point of Castilian literature (between mid-12th and early 13th C.), and Alfonso X of Castile ('The Wise') is often credited with initiating the process of seperating Castilian from Latin, and of Castilian becoming the official language of the kingdom Castile and León. Alfonso ordered large amounts of translations from Arabic, as well as the production of literary, legal, religious and scientific texts.
The first systematic grammar of a Romance vernacular was Antonio de Nebrija's Gramática castellana, published in 1492. He had been influenced in his wish to restore Latin through teaching the Romance by Italian thinkers like Dante and Lorenzo de Valla. While his grammar was not a success in its own time, other Spanish, French and Italian grammars followed in its wake, contributing to the emancipation of Romance languages from Latin.
Roger Wright is very good for differences between Latin and Romance, for example:
For a more general view on Latin and vernaculars see: