The simple definition of damping is to reduce oscillation. If we use a broader definition as in physics, it would be to reduce or dissipate vibration. I was making a distinction between different types of vibration control, but it does not matter. Critical damping is just enough to prevent vibration or enough to allow something to return to its rest position. Overdamping is a state where vibration is prevented to the extent that a moving part is prevented from movement.
A fluid damped tonearm does not have to have the fluid located at the pivot, for the pivot to be damped. Attach a paddle to the base of an armtube and have it dipped in a fluid trough when tracking a record, and you're damping the pivot. Overdamp the pivot and you're restricting arm movement.