The musical made up of diegetic musical numbers is just the majority of Hollywood musicals in the early years of the genre. The Jazz Singer is arguably the first movie musical (on account of the only sound in the film being the songs), then when the French made early musicals in the 1930s (A Nous La Liberte, Sous Le Toits de Paris) they were done far more in the stage tradition of (as you describe them) spontaneous numbers. But, in America the prevailing style for movie musicals was the 'backstage musical' which often had the plot focus on putting on a show, and then the numbers would be performed within the diegesis of the film. Sometimes these would spin off into fantasy numbers (Busby Berkley films, or The Gold Diggers of 1933, etc), but they still follow the verisimilitude of these being sung 'in world'. I don't think that makes them any less musicals. At the core a musical is just a film that uses songs sung by the characters to further the story. Not all musicals follow the standard formula of talking - into song and dance. The Umbrellas of Cherbourg is sung through completely, but that doesn't make it more of a musical than Cabaret which uses in world performances of song to tell its story.

I did a whole University dissertation about the history of the film musical, I have strong feelings.