Motor mover

It doesn't matter so long as you remember to disengage the motor mover before you try to drive away!

When we had a caravan we did both depending on the ground conditions. Not releasing before hitching means there is more likely hood of forgetting before diving off. However, depending on the slope releasing may result in a slight movement of the hitch meaning it is no longer aligned. In general my preference was to disengage before hitching.

Depending on the surface that the caravan is standing on, I find that I sometimes have to jiggle the caravan forwards and/or rearwards just a fraction for the towball to engage fully in the hitch and for that the motor mover is quite useful. That's why I normally disengage the motor mover after completing the hitching operation.

Likewise.

I've hitched then disengage the mover. 

I generally leave the motor mover engaged as I find it much easier to move the caravan small amounts to get it lined up with the tow ball. Also I’ve known the caravan to move/settle slightly after applying the handbrake and disengaging the mover - meaning you either re-engage it or use some muscle power.