Aaron, I will need to check my People 250 tomorrow, but it seems to me you should have at least 3 lugs for wire connectors on your solenoid. Two larger ones, for the heavier cables, and one smaller, which will run to the button for energizing the solenoid coil, enabling it to make contact internally for the heavy load of the starter.
Best to jump from the battery hot lug to the smaller lug, which will energize the unit as designed, though jumping the big ones will work, bringing pretty big arcs because that is the heaviest load your battery ever encounters !
The starter could be funky and drawing too much juice, also...
If you have the skill, cleaning internally and lubing could help.
redk