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.