The only way to really clarify this is to examine a thermostat and water pump together, but I still believe the the second disk is controlling a pressure relief bypass for the pump allowing water to short circuit straight back from the pump outlet to the inlet when it is open.
Pressure relief valves for oil pumps have been standard practice for decades, but as noted earlier they are positive displacement, so can generate very high pressures. For the water pump the problem is that excessive pressure drop means it "thrashes around in the water" which leads to cavitation (formation of bubbles) which as they collapse generate shock waves which can then cause erosion. This is a known problem with pumps and also propellors on ships etc (this effect is also something that is deliberately exploited in some manufacturing processes, but there the cavitation is usually created by other means!). I guess reducing pressure drop will also save a little power, but I doubt it's a big % of engine losses.
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