That's the variable-speed setup you have, which is the earlier version of what later became the 3-speed system. These variable or 3-speed systems have a complicated parking mechanism in which turning the switch to "off" sends reverse-polarity current to the motor, causing it to run backwards, which declutches a one-way clutch spring inside an eccentric nylon cam between the motor crank arm and the wiper transmission bar. This cam rotates, shortening (because it is eccentric) the overall linkage length, pulling the wiper arms down all the way to the bottom of the glass. Once they reach this position, a park switch mounted on the wiper motor driven gear and its cover plate break the circuit to the motor which stops turning until the driver switches it on again.
The 1-speed (early) or 2-speed (late) setup simply carries on when the driver switches it off until the wipers reach the bottom of their normal sweep, then they stop. There's no swivelling cam or one-way clutch spring or anything.
Sounds like you're not reliably getting that reverse-polarity current to your motor when you switch it off, either because your switch is faulty or because there's a break or point of high resistance (corrosion) in the wire. The park switch in the motor could also be stuck or faulty, and yes, wiper motor and switch must be very well grounded or stuff doesn't work right.
