Use a hot wire from the big terminal on the starter to the small lead going to the solenoid. If the starter engages and turns over the engine, then there is nothing wrong with your battery terminals. If it does not turn the engine, then you have a problem with your cables (maybe internally) or the battery negative ground lead connection where it attaches to the frame.
If the engine turns over when hot-wired, you have a problem with the key switch or start relay. They are not delivering enough current to the starter solenoid to make it fully engage. Be sure to also check your fuses. Fuse #2 is a common culprit on this model tractor. If it is blown, you will see the PTO light on the dash panel even though the PTO is off.
BTW: You should probably remove the black cowling under the steering wheel (both sides) and inspect for wire damage. Many folks have had rodents set up housekeeping in their tractors and chew wires. Also, remove your instrument panel (two screws) to get to the fuse assembly. Your tractor starting and dying sounds like a bad electrical connection or bad key switch that opened with vibration. That would cause the fuel solenoid to drop out and shut off the engine. Also a bad seat switch will cause the solenoid to drop out. You can easily bypass the seat switch by removing the connector and putting a wire between the two sockets in the connector. That will eliminate the seat switch as a culprit.