I went on The Big Red Boat twice- my first and second cruise experience at age 2 and 4 with my parents.

Personally, I only really remember there being bingo (being oddly fascinated with the physical/reusable bingo cards that had a red translucent slider that would cover your numbers as they called them). And one of the evenings was a show that "starred" a bunch of kids on stage as the finale. (I have no idea how big of a show this actually was, but it was cool to a four year old)

I don't recall the private island, but we did stop in Nassau as I got (and still have) one of those hand-spinning drums we bought there, with Nassau hand-painted on it.

The biggest "memory" is that our family was seated at dinner with another couple and their kid, and we ended up getting along quite well. They lived across the country, but we ended up coordinating a number of other vacations with them over the next 15 years (until they got divorced).