The last time I checked there were two cooling loops, ICE and MG/inverter package. I also understand that on the ICE, there is a 3 way valve and both electric and mechanical water pumps. The 3 way valve is used when the ICE replaces cold coolant with the warm coolant from the thermos at startup, while the engine is running to maintain the temperature of the coolant in the thermos by doing a coolant exchange while running, and at power down to put warm coolant into the thermos, unless the coolant in the thermos is already hot, or has recently been exchanged during a running exchange. The coolant system on the Prius is pretty dynamic.
Also, I understand there are 2 electric water pumps on the ICE, one used for thermos coolant exchange (the one you hear at startup and shutdown), and one that is used to cycle coolant through the engine when it is not running, but the vehicle is running and the engine has recently been deactivated by the ECU (while in stealth mode). This makes sense because it would be foolish to just store hot coolant in the ICE during off and on cycles. The temperature differentials created by that activity could be very damaging to the block as it might contribute to block warp. While underway and the ICE is running a belt driven water pump is in operation.
The MGs and inverter/converter use their own cooling loop, have their own isolated coolant supply (although same fluid as is used in the ICE) and pump. They also operate at a significantly lower temperature than the ICE. This makes sense because the ICE is burning fuel, however the MGs and inverter/converter are warm due to electrical resistance through the various electrical conversion processes. That generates a whole lot less heat than burning fuel.