I have only ever once cooled a cake dome side down. Now I know a lot of you use the waxed paper or parchment or paper towel levelling method and having that paper on top of the cooling rack probably makes a difference. But when the cake sits directly on the cooling rack as actually they were designed to do, well then what can happen is this. Your cake automatically wants to level out itself. So it tries to do so, so the cake basically tries to level and because it has that open space to go through, well this is how you can end up with a lump in the centre of your cake or a bulge. This is why the normal method of cooling a cake is to turn it out and then flip it over crown side up.
Incidentally the wax paper method of cooling goes back to the old Wilton encyclopedias.
Does cooling a cake on a rack alone make a difference? Yes it does. You end up with a cake that does not stick to your cake board, which I guess has plus and minuses. The big plus is that when you cut the cake, the outer part doesn't stick as readily to the board because the extra moisture isn't stuck there. Now some folks may find this a minus.
Flipping your cake directly onto your cake board can cause the board to warp somewhat. I have done that once in a while when I needed my cooling racks for something else, but only for smaller cakes as this can cause board warping issues.
The type of cooling rack you have is also important. Some are only a series of vertical lines and they don't support a cooling cake well, A grid pattern cooling rack like Wilton's it the best type. Most professionals will tell you to raise this rack up so that it has an inch or two underneath to allow for even cooling.
Personally I find a cake that cools on a rack, is a sturdier cake to work with.
I don't have any concerns about the moisture as I immediately level my cakes after they come out of the pan and then I brush on warmed thinned down apricot glaze. I put waxed paper under the cooling rack to catch the drips and yes, end up having to clean the cooling rack, but I think it does a better job this way. I also feel that levelling a cake with a serrated knife and levelling it so that the side is level with the top, not just levelling the top as many do, well this does a better job. And I don't find that any of the other levelling ways work as well, but that is just my opinion. We all have our own ways of doing things and I have tried them all.
I have cooled cookies and cakes directly on plates and such and you still end up with a good product, but you can see a difference when you compare the two and you can also see a slight texture variance.
If you wrap a cake before it is cool it will affect the texture somewhat, giving it a more dense and slightly rubbery consistency. Whatever you do, never wrap and freeze a hot cake as this is very dangerous and can compromise the other things that are in your freezer.
Hugs, Squirrelly Cakes