The terms used in this question are often unspecific due to the variety of contexts people will use them in. A linguistic answer would look at the problem in terms of first-language-acquisition (L1) or second-language-acquisition (L2). Both sociolinguistics and neurolinguistics can demonstrate how these are distinct processes with very different assimilation paths. L1s are passively assimilated in infancy and childhood, while L2s are assimilated with more difficulty and the active participation of the speaker. Academically speaking, it has been suggested a language is living if it still has a renewable source of L1 speakers. If the language has neither L1 nor L2 speakers, it is extinct. If it has nonrenewable L1 speakers (living elders but only L2 children), then it may be considered living and endangered or essentially dead depending on the analyst. But I would stress that this kind of distinction is relatively moot in both analysis and field research, whose primary goal is to record the language, not to preserve it.

Revitalization efforts, such as those in California for indigenous peoples who have lost their historical language in the Athabaskan or Uto-Aztecan families, rely on earlier records. No doubt there would be a strong tendency within these groups to suggest the language still lives simply because people are trying to acquire it, but there are logical complications to suggesting true continuity here. Recorded language is always an incomplete picture. For example even if you had a complete lexicon, and a thoroughly documented grammar, the instincts the original L1 speakers would have for forming new words, for acquiring loanwords, and for nonce-words or wordplay, are likely gone. The new L2 speakers may succeed in generating new L1 speakers if they then have children and isolate them in the L2 environment, but having skipped a generation, these L1 speakers will come to 'fill in the gaps' wherever they may exist in the imperfect L2 acquisition of their parents. Politically their speech may be identified as Yorik or Serrano (a few extinct languages in California) and have importance within the tribe, but academically this is not the same language. It may be more useful to look at such a case as a descendent language.

Take for example Hebrew. Hebrew does not enjoy a continuous chain of L1 speakers from Biblical/Ancient Hebrew. It is cited as successfully revitalized - often cited as the only such case - but modern L1 Hebrew speakers have pages and pages of their own innovations both lexically and grammatically that are influenced by other modern languages and do not descend from the structures used by the strictly liturgical Hebrew which was many people's L2, but nobody's L1. For example native Hebrew morphology follows a root-and-pattern structure like other Semitic languages, but in modern Hebrew a growing number of words do not adopt native patterns. This is common in areal linguistics even among 'unbroken' living languages - the morphology will often simplify or adopt innovations from neighboring languages of influence.

When this happens to a revitalized language, and it is inevitable that it will, the original teachers - the L2 speakers - will often perceive it as a threat and attempt to enforce conservative speech patterns on their L1 children. This always fails for the reason language change occurs at all - newer generations have their own innovations. In revitalized languages, they simply have more of them because their education could not provide for every possible situation.

Therefore there are ways in which the differences between a revitalized language and the original can be downplayed because the processes at work in those differences are common to all languages over time - but differences still remain that would make it a bad idea to consider the two stages of the language identical or continuous. That discontinuity may ruin revitalized speakers from participating in studies where the original language was relevant.

You will also find single instance cases of parents - for example - teaching their children Latin or Attic Greek as an L1. In every recorded instance of this, the child has reverted back to the language or languages used in their community in all contexts except home, and as more modern vocabulary demands to be incorporated into their speech - a process L2 Latin-speaking parents can't hope to keep up with - even the home context has reduced or disappeared. This fails for the same reason parents who teach their kids Klingon fail: the lack of peers. We ultimately learn language from our peers when peers are available, in preference to our parents. If you had a group or community of L2 Latin speakers who all agreed to speak only Latin with their children so as to produce L1 Latin speakers again, then at best you would have the Hebrew scenario. Is it or isn't it really Latin? How much innovation makes the language a distinct form? It'll certainly still be mutually intelligible, after all. These are not questions linguists can answer, because they have no real bearing on study. The old joke goes that a language is just a dialect with an army and a navy. And speakers of the same dialect do not always share the same features - usually they share anywhere from 20-80% of the same features (a specific set of features is then called an idiolect, the speech of one person). Intelligibility is often used to attempt to define a conventional language barrier, but when is a dialect a different dialect? There's always a town on the border, after all.