Of course, an aircraft carrier loaded up with planes, bombs, fuel, etc. will float lower in the water than an empty aircraft carrier. So it does make a difference, just not enough of a difference to sink the ship.
Here’s another way to look at it:
Imagine that you take a plastic bag and fill it partially full of water. If you put into a tub of water (and gently make sure the open top of the bag stays up), it will float at a level so that the level of water in the bag is the same as that outside the bag. In other words, the volume of water in the bag displaces the that same volume of water outside the bag. Put in yet other words, the weight of water in the bag displaces the weight of water outside the bag.
OK, now take out the bag, and replace the water in the bag with a hollow metal shape that is the identical size and weight as the water in the bag. If you put the bag full of a metal shape into the tub again, it should again float at the same level as the water and displace the same volume and mass as the water`outside.
The next step is to change the metal shape somewhat, while keeping the weight the same. Leave the shape the same below the “waterline” but extend it up higher. Assuming the weight remains the same, if you put it into the tub, it should float at the same waterline and displace the same weight of water.
The conceptual issue is that you are thinking of not starting with a bag of water and turning it into a boat, but taking a boat and putting it into the water. But, if you had a bag the size and shape of the boat, and filled it with water the weight of the boat, the bag would float at the level of the water in the boat, and displace that weight of water. Simply replace the water-filled bag with the actual boat, and there you have it.