So far we've only ever done one excursion through Disney. I highly recommend booking privately. Every single independent excursion we've done, we've gotten a head start before the Disney excursions, probably because of the time it takes to round everyone up. Not sure if this applies to every excursion, but it did on Grand Cayman in particular.
After a lot of research, I booked a private half day charter for our family of eleven through Native Way Watersports. They're highly rated on several sites including another cruise forum. This excursion was the highlight of our trip. Our driver met us at port and was easy to find as he was holding a sign up with the name of the company. He gave a very interesting narrative of the history of the island on the way As we got on the van, we noticed a very large group of passengers still waiting for the bus to pick them up for their Disney sponsored excursion. By booking privately, we were able to get to Stingray City ahead of the cruise ship groups Once we arrived, there were only three other small boats in the area. We also spent time at starfish point which the little ones in our group loved and were then taken to another site for some amazing snorkeling.
It was truly an amazing experience, our captain was engaging and great with the kids, and having the boat to ourselves was awesome. Oh, I haven't mentioned the stingrays yet. Beautiful! We were given squid to feed them and we were able to interact with them for a good while, separately from any of the other groups. Don't be afraid to book privately, tour operators don't want you to miss the ship any more than you do. With sites like Trip Advisor and social media, they don't want anything negative to be posted which could cause them to lose business which is their bread and butter.