just sitting on the bench, it does not turn by the power of the battery.
Sounds like a defective starter, but connect your voltmeter (both + and -) up directly at the starter while bench testing. If your voltage readings remain above 9 or 10 volts
under a load, you can
only then be certain your starter is defective.
But If your voltage readings fall below 9 or 10 volts under a load, you might just be fighting a weak/defective battery or high resistance/poor connection in your battery cables.
(these +/- battery cables would
always need to be 10-gauge or larger in size...)
If your not getting any solenoid click while testing on the mower, Its possible you might be fighting 2 separate problems here. You can use use a voltmeter to follow the starting circuit along its path and diagnose any problems it may have.
For a 4-terminal starter solenoid to engage, it would require:
Power/+ to 1 (either) of the of the small terminals,
Ground/- to the other small terminal,
Which would then finally engage the solenoid and make the connection between the 2 large posts and send power along it way to your starter.
Some solenoids also require a good ground source to they're mounting bracket.
A 3-terminal starter solenoid would function just the same, but just not have the additional small terminal requiring a ground source. But its mounting bracket would definitely need to be grounded for the solenoid to function.