Of course, because the cut is subjective.
That doesn't follow.Good, then it's not physics, and I can forget about it ;-).
Selecting a system to study is necessary for most of physics. But it is always a subjective act - since Nature isn't divided into system and the rest of the universe. Most of the time we don't want to consider the universe as a whole but only the particles in a particular beam, or particular materials, etc., and in each case it is the physicist that makes the selection, not Nature.
The cut, and the associated reduced description is of precisely this kind - specify precisely which system you are studying and you have specified the corresponding cut.