I believe there is usually only one thermal cutout in the circuit. If there were multiple cutouts they would be in series so any one tripping would cut the pump altogether. The two step shutdown seems to indicate the first click is the starter winding switch kicking in as the motor slows down, then the second click could be the thermal cutout. There is a centrifugal switch on the armature that will open once the motor comes up to speed, and conversely will close if it slows. I accidentally put 120V on a motor configured for 240 once. Under no load, such as when priming, the thing would spin up to speed, but when loaded, as when pumping water, it would slow and if loaded enough, would slow to the point the starter winding would kick in again. Then, back up to speed, slow down, and would repeat this cycle. I don't quite understand why the thermal cutout would trip unless it's a borderline trip point and being on the starter winding pulls enough current to trip it. Heating of the thermal cutout switch can be due to either high motor temperature or excessive and/or prolonged current drain.
Some questions:
1--When it first starts up is it running at full speed? You kind of indicated it is.
2--Any strange noises like it really loaded down? Loud hum etc.
3--Is the motor case hot to the touch?
4--If it's wired for 240V is there any possibility you lost one leg and are somehow applying only 120 to it?
5--Any unusual smell as in maybe a cooked winding?

Sorry...have no idea what type of pump is in the pics.
Hope this helps a little.
Al