Why use PI controller for speed, instead of full PID?
You do already have a D-term when control velocity. It is the differentiation of the position, so no need to do double differentiation - not stable.







simulate this circuit &ndash; Schematic created using CircuitLab

EDIT:

User DKNguyen has probably found a good reason why PI instead of PID is used:
Let' take for exapmle a real servo driver having 62.5us sampling time for current controller, 125us for speed controller and 1ms for position controller. Now, the most straightforward to tune is the current controller. You can compute the Kp, Ti directly from motor resistance and inductance. 

The velocity parameters are more tricky, the load and its inertia comes to play here. The algorithm for an integrator can suffer from precision while doing computation. Tiny increments could not be summed in a floating point number. But the real pain is to make a differentiation. In a book it was mentioned that sampling time has to be somewhere from 1/5 to 1/20 of dominant system time for PID controller to work as expected. Thus, high sample rate could spoil the PID, which in servo drive has a fixed sampling time.