OK, my 2p worth.
If we are in imminent danger of hitting someone else, I expect (UK at least) ATC to use the standard phrase "Avoiding Action".
If we are in the realms of preventing the above from happening at some point in the future, I expect to be told to reach / pass a certain level by a defined point in space or time.
If you simply tell me to expedite climb or descent I will work on the assumption that there is no potential for loss of seperation, but that a potential horizontal shortcut (for me or indeed someone else) is pending once the vertical situation is sorted.
CPB