Except for The Power of Blast! (which was a condensed version of Blast!), I wouldn't count most of the Disney theme park shows being adaptations of the actual broadway musicals.
Most of the time, the Disney theme park shows predate the Broadway musical adaptation. Both Aladdin - A Musical Spectacular, Frozen - Live at the Hyperion (Disney California Adventure), and Frozen - A Musical Spectacular (Disney Cruise Line) predate the Broadway musical adaptations, and are staged and told differently (both the Frozen adaptation at DCA and DCL still have the trolls rather than the HIdden Folk on Broadway, Iago is a puppet controlled by a performer on stage in the Aladdin DCA musical rather than just a performer on Broadway, "To Be Free" from the Aladdin DCA musical never made it to Broadway, etc.) And Disney Cruise Line's Beauty and the Beast appears to be inspired more by the live action film than the Broadway musical.
Still, it's been cool to see more Broadway style shows at the Disney parks and cruises, even if they are different than the actual Broadway musical. Actually, one of the original Elsas (Chelsea Emma Franko) from the Frozen DCA production is now currently the Elphaba standby on the Wicked National Tour. Of course, Eden Espinosa (who performed in DCA's first musical production Steps in Time) was the original Broadway standby for Elphaba.
Before I “discovered” my love of musicals I used to cruise. A lot. Did 50 cruises in total, lots of them in the Caribbean, Bahamas, five to Alaska, a handful to Canada/New England, several out of NYC, and once out of Baltimore, etc. One of those was on the Allure of the Seas out of Fort Lauderdale where I saw “CHICAGO”. That was in 2012. The last cruises I did was a 14nt back-to-back on Royal Princess in 2013.
Festival of the Lion King is more a celebration of the film rather than a adaptation of the stage musical. But it does have live singing by its talented performers, unlike the Frozen Sing-Along Celebration at Disney’s Hollywood Studios (and previously at Disney California Adventure till the Live at the Hyperion musical opened).
Festival of the Lion King (or FOLK, as it's called down here) was supposed to only last a single summer and was kind of thrown together at the last minute to give guests something else to do in the new park, which at the time was short on attractions. It became a massive hit with guests, and a couple of years ago moved to a new theater to accommodate the arrival of Pandora. The original production was in an outdoor theater, which was eventually enclosed once the show took off.
While the Frozen singalong features no live singing, it does feature a comedic duo of "Historians" who narrate the show. One duo, in particular, easily the best pair, has gained popularity outside of the show and has a (very funny) Youtube series.
Going back to the 90's--Norwegian Cruise Lines had condensed versions (about 75 minutes if I remember correctly) of Broadway musicals. I remember seeing Crazy for You and Smokey Joe's Cafe. I believe they also had versions of A Chorus Line, 42nd Street and others.
"Allure of the Seas out of Fort Lauderdale where I saw “CHICAGO”."
We saw that production in 2011 and it was actually stop-me-in-my-tracks amazing to watch! The design perfectly mimicked the Bway revival look, sure, but the dancing and acting was just staggeringly tight and on point. Whoever got that production up and on its feet should be commended for nailing the piece. And on a cruise ship that sleeps 5400 guests of all places-- who knew?!
Not a full production by any stretch, but I absolutely adore that Sunday Clothes/Before The Parade play on a nearly endless loop up and down Main Street at Disney Parks. It always brings a smile to my face and I'm glad Jerry gets probably a few hundred dollars a day from that alone. :)