Water heater confuses me; which thermostat?


I can't figure out which thermostat needs replacing, or why, or maybe it is some other part:
My Sears model 153.318560 52 Gal. Power Miser 8+ electric water heater (which looks an awful lot like an AO Smith heater) turned off and I had no hot water. I reset the heat limit switch of the top thermostat, and later I measured the water temp at 170 degrees. It turned off again. This time I had to reset both the top and bottom thermostats, and I later measured the temp at 180 degrees.
That second time, by watching the heater and the power meter outside, I saw that the red light, which means the top element is running, stayed on for 20 minutes or a little less, and that the bottom element was working. I don't think 20 minutes would be enough to raise the temp to 170 or 180, so I decided the bottom thermostat was bad and I replaced it.
Now** the red light doesn't even go on at all, but I watched closely and checked early and the temp got up to 160 before I turned it off.
The only thing left to replace is the top thermostat, but I'd like to understand how that could be the problem, or be told what the problem is before I replace it. Any ideas??
I checked for a short between the heating element and ground. There is none.
**After the bottom thermostat was out, but before the new one had come in, I wanted to take a bath, so I turned the water heater on with no bottom thermostat. The red light went on and I soon had enough hot water for 3 inches in the tub. But when I tried that the following day, when I had used all the hot water, nothing happened. The red light didnt' go on, and it didn't go on the following day after I had installed the new bottom thermostat. Did I break something by running without the bottom thermostat?? How would that be worse than running it with a broken bottom element? (the bottom element circuit is a simple series circuit with the thermostat and the heating element.)
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I don't see where you determined that you had power to the lower element, or even the lower thermostat. Typical wiring would have one leg of the circuit going to each element. The second leg of the circuit goes to the upper element via the upper thermostat. Once the upper thermostat is satisfied, it disconnects from the upper element and sends that leg to the lower thermostat, which is normally closed and sends the leg to the lower element. The lower element heats until its thermostat is satisfied or the upper thermostat calls for heat

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