Introducing hot water into a freezer rapidly raises the ambient temperature of the freezer. This causes the thermostat to click on, and the heat pump therefore runs until the ambient air drops down below the thermostat's shut off point. A lot of heat is removed very early on, and the ambient freezer temp is at its lowest going into the liquid to solid phase change.
With the cold water, the ambient air of the freezer doesn't rise nearly as much. The thermostat does n't click on until a long while later, once enough heat leaks out of the freezer. The ambient air is warmer going into the phase change.
It doesn't work if you put the two cups in the same freezer, but at least it could explain some anecdotes.