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It's not uncommon for a weak starter to turn over very slowly when the engine is hot. The fact that you're able to cool the starter with water and have it start tells you that it is pretty surely a starter issue.
There are two tests that you can do to confirm that the starter is the issue when it acts up.
First do a voltage drop test on the ground side of the system. With a digital voltmeter set to the 20v DC scale put one meter lead in the center of the battery negative post and take the other lead to an unpainted, grounded metal part on the engine. Hold the key in the crank position and read the meter, if you see more than about .5v then you have a poor ground to the engine, either at the connections or internal in the cable.
If this is ok then move to the positive side of the system.
If you do hear the solenoid click then you know the starter should be engaging. Again leave the meter set to 20v DC, put one lead in the center of the battery positive post. Take the other lead to the positive input cable at the solenoid, hold the key in the crank position. If you see under .5v then there is acceptable voltage drop between the battery and solenoid.
Then move one lead to each of the positive connections on the solenoid and do the same test, if you see more than .5v then replace the solenoid.
If that's ok then take one lead either from the positive battery post or from the solenoid and take the other end to the starter. With the key held in the crank position again check the voltage, if it's less than .5v and the ground side was also ok then replace the starter.