[142 Guy]

To be clear, when tested off the car the starter motor starts up when you energize the 'trigger contact' and stops when you de energize the trigger contact?

As Ron suggests, I think the problem is related to the solenoid contact operation; but, since it worked OK off the car; but, not on the car I think the 'cause' may be the start drive attached to the solenoid (the Bendix 'thing'). Typically you energize the solenoid which simultaneously closes the main contacts to the starter and engages the starter pinion gear with the ring gear on the flywheel. It seems like when you de energize the solenoid the starter pinion is not unable to retract from the ring gear which is preventing the solenoid from opening the main contacts keeping the starter energized. It is not until you disconnect the battery that the pinion is free to retract and the solenoid armature moves to the open position.

With the starter removed from the car, check to make sure that the armature of the solenoid is free to move back and forth and also check the fork / actuating arm to make sure that it moves freely. Finally, examine the actual starter drive to look for signs of damage - not exactly sure what that would be because I have never had a problem with the starter drive.