No heat on idling, occasional overheat light on/off

I started having some heating issues on 2010 Prius. Occasionally when the car is completely warmed up (20+ mins driving) on idle (long stop sign, parked with car running) vents start blowing colder and colder air until it's just cold. When I start driving - all goes to normal. Also there were two times (today was second) when coolant overheat light pops up. I park, turn off the car, wait couple of minutes - it goes away for good, driving after that for 30+ mins, doesn't come back. Engine(!) coolant level is on full and doesn't leak (don't even remember when the last time I refilled it). Water pump on Prius is electric, so if the problem would be in it, I assume it would throw some code on me, but there is no MIL or any codes stored. Thermostat might be the issue, which make sense about losing heat while idling (coolant runs on long circle, therefore doesn't warm up well enough to keep the heat), but overheat light don't make sense in this case if the "stuck open" thermostat. Anyone any suggestions? Or maybe someone had similar issue with their Prius? So far the only logical explanation for me is "glitching" thermostat that gets stuck sometimes in "open" position, sometimes in "closed". So I call for wisdom of Prius Gurus on this forum to identify the issue

 

Around 200K on a clock, second owner, got it at 180K 5 years ago (my parent's-in-law car, so they don't drive it much). First owner was religious with capital "R" about maintenance on this car, kept literally all records from day one, but there are no records about water pump whatsoever, so I assume he never had issues with it.

After talking in details with dad-in-law (he had second occurrence of overheat light) I found out that heat disappeared while driving and after another ~5min of driving he also got "overheat light" on. By the time he made a turn to a parking lot and prepared to stop and shot off the engine - light went off and heat came back after short period of time. He kept driving and light never came back and heater worked normally. Which eliminates the thermostat version completely (that is not how it fails) and give more reasons to believe that intermittently coolant stopped circulating in the system, which probably caused by stuck water pump impeller (considering the mileage on this car). Looks like I'll be on a hunt for a new water pump. I'll update you after pulling the old one to see if we were right. Thanks.