At what point a language changes so much that it becomes another language?
That is a difficult question, that probably can only be answered in hindsight. At no moment in the convoluted history of Romance languages, anyone woke up and realized, "gee, this thing I am speaking is not Latin, it is, wow, French". People just spoke the language of their time and place, or one or more of the languages of their time and place (in France, likely, Vulgar Latin and Frankish).
At that time in history, people certainly distinguished Latin - the language in which the priest said the mass, which was mostly incomprehensible to them - from the language they actually spoke, which was, for lack of a better word, generally called "Vulgar" or "Vernacular", and quite certainly didn't constitute a unified language - it was different in Lisbon or Catania, Nice or Santander. When in contact with people of different places, people would recognise that those from other places spoke a different language - sometimes so far apart as to be unintelligible, sometimes close enough to be described like, "those damn Genoese and their ridiculous and wrong way of speaking". But contact with foreigners was rare, and probably most accessible to people who could manage some Latin - Medieval Latin, already quite different from Classic Latin, but that remained the language of high culture throughout Europe during the Middle Ages.
When the Romans conquered Western Europe, there were already several differences between the language spoken by the Roman elite and the Roman populace, to which most of the conquering soldiers belonged. It is this variant of Latin - Vulgar Latin - that originated the several Romance languages (with the exception, I think, of Sardinian, which is more related to Classic Latin). To the extent that Romans were conscious of this dichotomy between their standard language and the language effectively spoken by common people, the divide is very old. Ciceron probably knew his language was different from the language of his servants, though he probably framed this as "proper Latin" versus "wrong", or "debased", Latin.
Vulgar Latin then evolved differently at the different parts of the Roman Empire/influence area. At some point in history, the differences were big enough that we can no longer speak of "Vulgar Latin" as one language. Unfortunately, this was mostly an undocumented process: people actually changing the language were illiterate, so they didn't write their language; and literate people wrote in Medieval Latin, not in Vulgar, and weren't really interested in writing about the commoners' language(s). But at some point the language spoken in the Iberic Peninsula was Iberic Romance rather than Vulgar Latin.
Apart from the scant documentation, we can use comparative method to understand such changes.
For instance, here we have the history of a single word, from Classic Latin to Modern Portuguese:
Classic Latin: genu- (/genu/)
Classic Latin (diminutive of the former): genuculu- (/genukulu/)
Vulgar Latin: genuclu (/ʒenuklu/)
Iberic Romance: *genollo (/ʒenoʎo/)
Old Portuguese: jeolho (/ʒeoʎo/)
Modern Portuguese: joelho (/ʒoeʎu/ or /ʒweʎu/))
How do we know that the order of the changes is that, first /g/ to /ʒ/, then /kl/ to /ʎ/?
Apart from documentation (we have Old Portuguese texts with the word "jeolho", for instance), we know that Medieval Latin has /ʒ/, and not /g/, so this sound must have changed very early and was probably done in Vulgar Latin; we know that all Romance languages have changed /kl/ into something different, so that it must have happened later, when Vulgar Latin was morphing into several different Romances; we know that Portuguese alone dropped intervocallic "n" (compare Portuguese "geral" to Spanish "general", so that must have been even later, when Iberic Romance was dividing into Galician-Portuguese and Castillian.
This process then intertwined with the political process of the formation and unification of modern nation-States. At this point, the local elites of Portugal, Spain, France, decided that they would be better served by making their local languages literate than by trying to repopularise Latin, as it would help in constructing the borders, and in creating a national ideology that could justify the existence of the nation-State. At that point we have the earliest literature in those languages. This process was unfortunate for several Romance languages that didn't correspond to the political boundaries - Mirandese, Occitan, Catalan, Leonese, etc. And it originates the saying, "a language is a dialect with an army", from which it is often deduced that the differences between languages is more political than linguistic (but notice, the army, and the State in general, are, besides obviously political entities, also glossopoietic) entities: military terms are coined by the army (and navy), and courts, police, universities, public schools, also coin a huge vocabulary).
In the specific case of Italy, the political process of unification of the nation-State happened much later, only in the 19th century, which favoured the permanence of several different local languages - Venetian, Sardinian, Sicilian, Tuscan, Neapolitan, Emilian, Lombard, Piedmontese, Calabrian, etc. - not all of which even belong to the same Romance branch (Sardinian is Insular Romance, Venetian is Western Romance, while most of the others are Southern Romance). So things in Italy are even more complicated than in Portugal, France, or Spain: while those have had national languages from the end of the Middle Ages, Italy chose Tuscan its national language, or based its national language on Tuscan (mainly due to the literary and cultural importance of Dante), only in the 19th century, much after everyone in the peninsula was very sure that what they spoke was not Latin.
So, at what point we say "These people in the Italian peninsula are no longer speaking Latin; they are speaking Italian"? At no point. They stopped speaking Classic Latin at the end of the Antiquity, when cultured Latin evolved into Late Latin (which in turn evolved into Medieval Latin later), while popular Latin evolved into Vulgar Latin. At the Middel Ages, they were speaking several different languages, which can be linguistically grouped into three different "families", Western Romance in the North, Insular Romance in Sardinia, and Southern Romance in most of the Peninsula and in Sicily. "Italian" as we know it was one of these Southern Romance languages, namely Tuscan, and spoken by a very small minority of the population up to the 1860, when Italy became a unified nation-State, and its State deliberately choose it a the national language, and enforced via public education (and active repression of the local languages, which became collectively know as "dialects").
As for "the split between Spanish and Portuguese" (which is really the split between Castillian and Galician-Portuguese), it happened in the High Middle Ages, certainly before the split the Condado Portucalense and the Kingdom of Spain. But then the political split reinforced the linguistic divide, making the languages further evolve apart from each other (and sparing Portuguese the fate of the minoritary languages of Spain).
And "at what point does Puertoriqueño cease to be a dialect of Spanish, and take on its own identity as a language? Similarly for Brazilian vs Portuguese, or the other various dialects of Spanish in Latin America, or even English/England vs English/Australia vs English/North America?"
At what point in time it is impossible to know. Modern communication doesn't favour the division of languages - and, so, the process is bound to be slower, possibly much slower, than the split of Latin into several languages. Unless, of course, there is a collapse of modern civilisation, that breaks the communication system apart, so that people in Puerto Rico, Brazil, or Australia, are no longer in touch with Spain, Portugal, or England (which, as a second thought, is what happened to the Roman Empire too, so the odds are, if there is a civilisation collapse, in half a millenium, when finally Venezolans rediscover Spain, people in Latin America will be unable to understand Spaniards, and conversely). At what point in linguistic terms, when the spoken varieties of those languages become mutually incomprehensible (text is more conservative, and may allow for the fiction of linguistic unity, perhaps useful for political purposes).