When you read the article about extinct language on the English language Wikipedia, it is defined as a language that does not have living native speakers. It is contrasted with a dead language, that is a language has stopped changing in grammar, vocabulary, and the complete meaning of a sentence. Living, extinct and dead languages are a range.
A language like Latin does not fit in one or the other category. Latin is, according to this definition, extinct. However, it is not a dead language as Latin has continuously been spoken in the Roman Catholic Church and consequently has evolved over the ages.
There are two ways out of this dilemma, the understanding of native, or changing the definition. One definition of native is "characteristic of or relating to people inhabiting a region from the beginning". With this definition Latin is not an extinct language as Latin has continuously been spoken in the Vatican. What makes this understanding feasible is that the word "mother tongue" is not part of the consideration. This is a departure of the understanding in the article where Latin is explicitly considered an extinct language. However, the examples of extinct languages is highly optimistic because the majority of these languages are likely to be dead languages and this makes the article less then authoritative in understanding the issue.
For me there is a practical application to this. When a language is dead, when it is no longer an evolving language, there is no longer a point to create a user interface for computer applications. There is no longer a point to create a new encyclopedia as an encyclopedia is to encompass all existing knowledge.
It would be helpful when an authoritative source would make this distinction between extinct and dead for languages with this express understanding of positioning languages on this scale from living to dead.
A practical question, is the historic Ottoman Turkish language (ota) living, dead or extinct.