"expedite climb" or "expedite descend" are often used to ensure a good (controllers point of view) rate of climb/descend and not to have it increased.
Expedite climb may soun like increase rate of climb but often means continue climb as you're now.
On radar screens the ATCo can read the Mode C and the update depends on the system and on the antenna speed (rpm). Some systems also show the average rate of climb/descend.
Speaking for myself, I often use the terms "expedite climb" with "until FL..." to let the pilot know about my need to have vertical separation at a specific altitude.