A private reply I got from a friend...

"When a starter continues to run with ignition key off, the problem is the contactors in the stater. These are two copper discs that allow battery current to flow to the carbon brushes.



The copper contacts weld together due the amperage loading heat.....like a welding rod.


They are made from copper so this welding should not happen .



It is likely the return spring of the solenoid is weak.....or solenoid plunger hanging up more than it should in it's bore.



DO NOT GREASE THE BENDIX OR CARRIER SHAFT. Dirt will case it it seize. Only grease the teeth of the flywheel."


Indeed, sounds like the contacts are welding. And that, they aren't made from copper like they are supposed to be. SO, I will be taking the sucker down soon and getting it properly rebuilt.

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1998 VW Cabrio
2001 Audi SomeRoad A/T (no air suspension)
2003 Audi DeadRoad M/T (no vroom, for later)
2002 Audi NoRoad A/T (nothing under the hood, being rebuilt)