When a battery cannot carry a load, thats the exact thing that happens. The engine will briefly turn over then the clicking starts. It happens everytime a batter goes bad. Voltage on battery means nothing. Its voltage while under a load, such as trying to start, that matters.
Sorry to tell you this, but that old thing of smearing grease on battery contacts regardless of which contact—battery post, ground at frame, or starter connection—introduced moisture into the connection and starts corrosion. Clean that mess off then use a degreaser or brake clean on all parts. Keep the connections clean and do a regular close visual inspection.
I have not read ALL the replies, but don't overlook the battery cables. Corrosion can occur in the insulated jacket and the cable will look fine but create a scenario that points to the solenoid. you can use one side of jumper cables to connect the positive to the battery to the power of the starter. Get on machine and start as normal so as to activate all safety switches. If fixed, either replace cable or cut the end off and reinstall new battery clamp on the end. The fault is usually in the first half inch of battery end of cable. If not solved, by hand, unhook battery cable from starter / relay, and with tractor IN NEUTRAL or an operator on machine with clutch in and out of gear, touch the jumper cable to the starter pole side of solenoid and the machine should turn over instantly. This will tell you that the issue is on the solenoid portion of the circuit. I guess you might also check the battery by jumping the normal way or from a battery charger as well. If this solves the issue than the battery most likely is the culprit.
I recently had a similar problem and replaced starter, solenoid, battery and battery cables which didn't help but then I found a loose ground wire hiding behind the engine and front loader frame that just came apart when I touched it. Replaced it and now problem fixed. Starts and runs better than ever.
