You didn't post a conclusion. If you jumped it once and didn't jump again, that's not the solenoid. As Kabota owner Mark said, that's low voltage.
You turn the key or push the button. A couple hundred milliamperes run down the wire to the small stud(s) and pulls the solenoid rod down making contact between the two ⅜" studs with the copper disc attached to the end of the rod. That action connects the starter across the battery which loads the battery and supply cables/connections.
Once the starter starts sucking on the battery it craters and either the terminal voltage drops or somewhere in the supply cables/connections, you have a corroded terminal dropping excessive voltage....but on this one I'm taking a sulphated up battery that can't supply a good load and maintain terminal voltage.
The voltage available for the solenoid to remain energized has now dropped below 9v or whatever the holding voltage happens to be. The lack of energizing current causes the spring loaded rod to release which releases the contact between the input and output terminals of the solenoid.
This takes the load off the battery and the voltage pops back up.....fingers are still on the start switch so you have adequate voltage back on the solenoid and it energizes again, loads up again, releases again, etc, etc, etc.
Have your battery load tested for 11v at the terminals while delivering 200 amps. If it won't do that get a new one.
Pitted contacts reduce the surface area of the contact surfaces and that reduces the current carrying capacity to the starter. It does just the opposite of what I said above because the starter can't load the battery supply like it should since it can't conduct adequate current through the solenoid.
Pitted contacts continue to erode until the surface area is so small that the copper to copper plate to stud junction welds together and this condition is where you let off the start switch but the starter continues to run. Then it becomes a.......geez, how do I shut this thing off.
