Since you didn't say your skill level, I'll take it you are intermediate at least.
Cozumel, I went with Christi's operation, BlueXTSea and recommend you do the same. Quality service, small fast boat, and cruiser friendly.
Grand Cayman, I went with Neptunesdivers and also recommend them. 8 divers max, follow the guide or do your own dive, and also cruiser friendly.
Make reservations as soon as you can since these smaller operations fill up early.
While I usually recommend making your own arrangements, in Belize I would probably just go with the ship's excursion just because of the logistics.
Jamaica and "good diving" is an oxymoron. I have no recommendation for this stop. Maybe the river tubing. I saw a trip report where one couple did this and it looked kinda fun.