Right now Latin has little place because of how the language system is implemented. If the benefice, like the halved oppinion, of sharing a language didn't only apply to the mother tongue of a character and applied to any language that two characters share knowledge Latin (or greek, arabic, etc) would have their place. That way learning Latin would be as useful as it was for Europeans in the era, as it would apply to churchmen and other cultured folk in Europe. However, I don't think that it should be learned from a character as it works in the present build, but a character with enough Learning should be able to learn it by himself, maybe via a decision to "Study Latin literature".
I would not use "Church Latin" to refer to "Classical Latin", nor "Classical Latin" to refer to "Church Latin". I think the game would do fine with just "Ecclesiastical Latin" as a language, but it should be noted that this is already different from the way Latin was written (and, especially, spoken) during the Classical period.
What constitutes Literary Latin depends on when and where the book you're talking about was written.
I would call it just "Latin". Ecclesiastical Latin isn't a monolithical entity and it changes in style according to the Latin Language Era (Medieval, New Latin, Contemporary). Also Classical Latin and Ecclesiastical are not that different, they differ in style, some vocabulary and pronunciation (which was "restored" quite after the Middle Ages), but in all they are the same language. A person who has learned Church Latin can perfectly read Classical Latin, apart from looking for the meaning of a word or two.
