The subtle difference in your quoted case between the QRH and MEL is that the QRH lets you carry on at any chosen altitude while of course avoiding icing etc. The assumption is that you started out with the whole system serviceable and suffered a failure or fault after dispatch.
The further assumption is probably that you would be very unlucky to have a further fault or failure in the system, thus they don't really expect that you are going to end up depressurized as the result of a pack or bleed trip. Same reasoning lets twin engine aeroplanes fly - i.e. they only ever expect one engine to fail on any one flight!
The MEL however, is intended to be used to get you away legally, that is dispatched, with one system already failed (obviously nothing as radical as an engine!). To satisfy the certifying authorities the MEL is therefore more conservative in that it has to allow for the possibility that the other system will fail or fault along the way. Also because MEL/DDG references usually form part of the planning process, it gives you the heavy hint to replan at the lower flight levels for fuel, time etc. How your company views 'dispatch' in this situation should be spelled out in the preamble to the MEL and/or ops manuals. It could be from commencement of pushback, commencement of taxy or commencement of takeoff roll, or it might be left entirely to commonsense in that there is no point in dispatching from a maintenance base under the MEL if it then grounds the aeroplane at an outstation.