The point is that people never stopped speaking Latin (although in some places like North Africa nad the Balkans they did) There is a clear continuity between Latin and various Romance languages. Some time during the early middle ages, people started complaining they couldn't understand such and such person's Latin and the different dialects eventually became to be called different langauges and had their own writing systems created.

Langauges slowly change over time. It's kind of like the Canterbury tales' English is different from Shakespeare's English and even Lovecraft's English is clearly more oldfashioned compared to modern English.

Shakespearian English and modern English are mutually intelligible and their separation is mostly temporal, not geographical.

Middle-English is referred to as a distinct language which you can study.

"Distinct" is pushing it a little. Compare the Canterbury Tales with original wording but modern spelling here ( http://academic.brooklyn.cuny.edu/webcore/murphy/canterbury/ )

    But first I make a protestation
    That I am drunk; I know it by my sound
    And therefore, if that I misspeak or say,
    Wit it the ale of Southwark, I you pray
    For I will tell a legend and a life
    Both of a carpenter and of his wife

The spelling isn't even that different (though the pronunciation is more so); compare the fifteenth-century spelling from www.chaucermss.org (multitext edition):

    But first I make a protestacioun
    That I am dronke / I knowe it by my sown
    And therfore / if þ I mysspeke / or seye
    Wite it / the ale of Southwerk I preye
    For I wol telle a legende and a lyf
    Bothe of a Carpenter / and of his wyf

(The slashes appear in the fifteenth-century text; I'm not sure what they indicate.)

Modern English and Middle English aren't quite mutually intelligible, but if you were somehow dropped into England in 1420, you'd probably be conversational inside of a month.

Mutual intelligibility is rarely the only reason for calling a different langauge different. E.g the scandinavian langauges are relatively easily mutually intelligible, so is Czech and Slovak, yet they are called different languages mainly for political reasons. On the other hand, many of the Chinese "dialects" are not even alike, yet they are still considered one language for political reasons. Even perhaps more familiar "Black English" is usually considered to be nothing more than English with bad grammar, even though it's very clearly a distinct language of its own, that is not readily mutually intelligible with standard american English.

You just have a quirky understanding what a language actually is. Language names are given and defined in language themselves, and it does make a world of a difference if you say you speak Swedish, Danish, or Norwegian, even though they are quite MI among each other, and even somewhat MI with German.

Linguists do refer to different languages in China, which are referred to by the government as dialects. So there is much more to it than politics. In your definition probably the whole Indo-Germanic family from Norwegian to Farsi or Hindi would be one language.

Romance languages are hardly mutually intelligible with latin...

So yes, latin pretty much died out as a language in the strict sense.

Endangered languages are considered to be dying out in a meaningful way, by losing their speakers.

All languages are equally "endangered", if you mean dying in the sense that Latin did, where it was and still is one of the most successful languages of all time.

Latin is not spoken natively anymore except for really rare special circumstances (much like Esperanto and Klingon have occasionally been taught/picked up by young children).

That is a useless definition, since all languages are constantly getting extinct that way.