I think it just comes down to a bit of training. (Great Teacher Quorum activity!) When I was a Teacher, the standard practice was to fill the cups under the faucet with med-low pressure (to avoid the obvious back-splash) until they were nearly full. Then the trick was to tip the tray about 30°, which uniformly emptied them down to an acceptable level, then wipe off the outside of the tray. Piece of cake.
You might be interested in some additional tips from
THIS similar thread from about 2 years ago.