Except that prior to 1492, there was no language known as "ladino." It was simply Spanish, and the Jews who were exiled continued to preserve the language. Does the community auto-identify with the term "ladino"? Or was this term imposed after the exile?

More accurate to say that at the time, there was no language known as "Spanish" - there were speakers of Latin dialects (including what became Spanish and Portuguese and Galician and Catalan &c.), speakers of Arabic, and speakers of Basque.

We are talking about 1492 here, there was already a established Spanish language

That same year this was written:

https://en.wikipedia.org/wiki/Gram%C3%A1tica_de_la_lengua_ca...

The presence of lengua castellana instead of lengua española in the title is exactly GP's point. Remember that Castile and Aragon only just entered personal union at the time! Prior to Spanish unification, Castilian and Aragonese were seen as different from each other as modern people see Spanish and Portuguese (and Aragonese is just as different from Castilian as Portuguese is).

In addition to Castilian and Aragonese, you also had, among others, the Galician language, the Asturian and Leonese languages, and of course in the once-Muslim lands not only Ladino but also Mozarabic (which parallels Ladino in a few ways: Ladino is a close relative of Castilian spoken by Jews in Muslim lands, while Mozarabic is a close relative of Aragonese spoken by Christians in Muslim lands).

The Spanish language is still commonly called "castellano" today.

Ladino comes from the word Latin and according to Wikipedia used to be the term for literary Spanish

Actually, it is more frequently called judeo-español (Jewish-Spanish) in Spain.