In terms of library music, "cutdowns" are simply shorter versions of the same composition. So if a full track is say 2:30 long, libraries will typically want a 30-second and a 60-second version of the same piece. This makes it easy for clients to drop the music into a commercial or promo, which typically run no more than 60 seconds. 10-second and 15-second cutdowns are also sometimes requested.
Each cutdown version should be its own standalone work, so ideally, it'll have a beginning, middle, and end. Obviously that's not always possible because of the short length, but it should at least feel complete on its own.
I think this same principle is being applied to the loops in Hammers. For example, the first key in a loop group plays a full 8-bar loop, but the remaining keys in that group only play 1 or 2-bar slices of that loop. At least that's how it seems to me.
Hope that helps.