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I lather right off the puck also. I just close the lid when I'm done shaving. Some times I rinse the puck off and some times I don't. I don't shave daily plus I rotate soaps so they don't get a chance to over hydrate or dry out.
Tabac and MWF are in glass jars with lids that simply rest on the top and are not air tight; I assumed this was to allow the puck to dry between uses. Old Spice is in mugs which have loose fitting lids, if they have lids at all. I suspect those soaps are intended to be used regularly, and not part of a large rotation of soaps only used every few weeks or months. If I were going to seal a soap for a lengthy period then I would ensure it was dry and no moisture was present in the container, 24 hours would be plenty. I might also question if I have too many soaps
I leave mine open all the time, but I usually only have two pucks going and I'm not that interested in the scent (one is unscented and the other would be if they made that option). I'm a daily shaver. I will say that the WK Pasha's Pride still retains scent, although probably not as strong.
I did over-hydrate MdC years ago by covering it while still wet. It didn't damage the soap, but certainly changed the way it loaded.
My best guess is it probably doesn't matter too much either way.