The benefit is in the Hot water
The cost of running a FL machine with cold water is negligible, the motors are small in comparison to a TL machine.
The saving comes from the fact, that when washing in 60deg C water, you're only heating about 10-15 litres instead of 80L.
When they first introduced FL machines with Energy labels in AU, the FL machines were shown to use more electricity than the TL machines. This wasnt taking into account that the hot water for the TL machine was being heated elsewhere. At that stage, the cost of running was based on the consumption of the machine, not on the total cost.
Now the labels show the energy use for both a cold and a warm wash, and it takes into account the cost of heating the water be it in the Machine or in the water tank.
In my Pre Miele days, I used to only wash business shirts in hot water, and everything else in cold.
Since getting the Miele, our water bill is down to 400L/day (From 900L/day) for the household, and I now wash everything at 40deg C except for towls and business shirts which I wash at 60deg C, our power bill is about $80/quater cheaper than it was previously.
The savings come from the reduced cost of heating the water.