The phraseology "expedite climb" doesn't mean anything.
Exactly.
Where I'm working we tell the trainees not to use the term, because it is not specific and open to speculation (as above).
Would be like saying: "Reduce your speed a little bit."
If it is urgent then we use the term "immediately" plus TFC info and/or "max ROC/ROD" (which would make it sound REALLY urgent).