Ok, both Cayman and Belize are tender ports and will continue to be so. In both cases the tendering services are fairly effecient and rapid.
On Cayman the ships anchor very close to the port and you can even see the people on the dock from your Carnival cruise ship. It is possible that some scuba excursions are right off the ship but if there is I haven't seen that or heard of it. On the Conquest in Feb 2004 we were turned away from Cayman because the swell was too much for safety. This November saw NCL ships using their own tenders which are their larger enclosed life boats but not aware of Carnival doing that.
Belize is somewhat of a different story. The water around Belize City is very shallow so the ship anchors several miles off shore. Some of the snorkle, scuba and trips to cays excursions are right off the ship but not all. For land excursions the tendering service to Belize City is used. It is a professional and rapid service using rather small tenders that use big yamaha outboards to get you there quick. Last trip there it was about 20 minutes each way and actually a fun boat ride (FAST). Most of the land excursions do allow time for shopping and knocking around and drinking at the tender terminal in Belize city.
Tender ports do take additional time off of your port time, and should be a consideration in your planning.
Cozumel which really isnt the topic is a different story but addressed elsewhere on this message board.