I think you are right and wrong. A double acting cylinder has fluid on both sides, I'm not 100% sure if it's equal on both sides (that may account for the rise and fall of the fluid). I believe the biggest reason for the added capacity of a reservoir is expansion. Previous to the hydraulic assist there was x amount of fluid in the system that expands, now there is y amount of fluid in the system that is expanding as it gets heated. Because it expands in direct proportion (i.e. same fluid), the more fluid in the system the larger the amount of expansion, and the greater need for a "reservoir". A good cooling system will sometimes work as it adds capacity and cooling, but if the fluid heats enough, it will need room to go.
An example would be that if a fluid at 70 degrees would expand 10% at 100 degrees, a system that started with a volume of 50L would now take up the volume of 55L, where as a system that started with 100L would now take up the volume of 110L. More fluid = more expansion at the same operating temp.
Just me thinking out loud......