It looks like those bouys are “mooring bouys”, which are floats attached by chain or rope to an anchor or other type of ground tackle, making essentially a fixed point for anchorage. Vessels will tie up to a mooring in lieu of using their own anchors.
Anchors for moorings use similar principals to working vessel anchors, but there are some differences. More commonly the anchor will be one or more mushroom anchors, which can be set deep into the seabed given some time, but are very difficult to pull out after they are set. Sometimes an anchor of simply very heavy weight is used, which can be quite effective if sunk into sand or soft ground. A vessel’s anchor has to balance holding strength, weight, and ease of storage and use. A fixed mooring anchor has holding strength as its primary criterion, and is usually more able to handle vertical and diagonal loads than a vessel’s working anchor.
Moorings typically use much less scope than a vessel’s anchor, both because the anchor is proportionately stronger and because mooring areas usually have limited room for vessels to swing. The cruise ship in the picture was moored fore and aft (moorings attached to both the front and back of the ship), which means that it is quite tightly fixed into place.
That is called Dynamic Positioning. Back when I was an oilfield supply vessel captain, DP systems were becoming common on OSVs and standby vessels. It is a blessing for platform supply captains, where you are required to hold the vessel in position for hours while the rig crane offloads and backloads cargo. You get in position next to the platform and press a button, and it holds the boat remarkably still even in rough seas. Beats the hell out of working the throttles manually while staring at the legs of the rig for hours.
I assume the mooring buoys might have been more necessary in Labadee because it is a fairly small bay, as opposed to in Georgetown where the ship was basically out in “open” water? As I said, I don’t think the buoys are used at all out in Georgetown (there was even a fifth ship out there, Norwegian cruise lines, I think, behind the Mariner).
The ship would not need moorings in addition to DP, the DP system would be capable of holding the ship’s position within a couple of feet regardless of the size of the bay. My guess is that they did not use DP in Labadee due to shallow water (which would also make it difficult to anchor). The ship’s thrusters would probably stir up the bottom in a shallow bay, damaging seagrass beds and clouding the water.
Those are beautiful ships. Enjoy your cruise, that sounds like fun!