ggersten said: "Do bio-musicals like Beautiful and Jersey Boys really count as jukebox musicals? I don't think so - as I think of a jukebox musical where pre-existing songs are shoehorned into a new story where the songs purportedly advance the plot or character. The "and then I wrote/performed this song" is not that show. If just having pre-existing songs in a show is "jukebox" than various revues should count. But on one has suggested "Aint Misbehavin'" (I've only seen parts of the televised version, so I can't judge it) or any of the plethora of Sondheim revues.
You raise a valid question; I would argue that the shows still don't have original songs, so you are not getting the opportunity to hear new music. Re Ain't Misbehavin', I didn't consider that, because it was a revue; there was no attempt to shoe-horn songs into a script.
So I guess there are really four types of musicals in this broad category:
-- True juke box musical, in which the songs are squeezed into a script in an attempt to propel the show along the same as an original song written specifically for that moment in the show would do, e.g., Mamma Mia, We Will Rock You, the musical version of the play and film Holiday several decades ago (I can't remember its name)
-- Live version of an original movie musical or concept album, probably incorporating a few more numbers, e.g., Saturday Night Fever, Flashdance, Footloose as well as American Idiot, Tommy
-- Show biz stories, which include songs associated with the original singer / group, played as performances, rather than used to propel the story, e.g., Jersey Boys, Beautiful, (Mostly) On Your Feet, (Mostly) Motown (I didn't see it, but have the impression that not all numbers were 'performances'.
-- Revues, which make no attempt to incorporate a script, e.g., Ain't Misbehavin'
So, where does a show like Crazy For You fit? Or An American In Paris? Probably overranalyzing it.
In reality, the previous answers seem to have determined that it is a book musical with songs that are not new.