Troubleshooting Toro Riding Mower Electrical Problems?

I have a Toro riding mower. The battery is OK. It had a loose connection which had interfered with getting power to the solenoid. I wriggled the wire a bit and the solenoid began clicking. After a few tries the starter engaged briefly (a second) and disengaged immediately.

Is the starter motor connected directly from the solenoid to the starter motor? Could the loose connection which stopped the solenoid operating cause the starter problem as well? Is the starter mounted on the outside of the engine and does it look like an auto generator but smaller? The gear from that "generator-looking" item is under a small cover and when I took a look the small gear was disengaged from the fly-wheel.

By Art