Well it's been a minute (or closer to a dozen years) since I messed with a wiper motor and linkage. If the wipers aren't clocking themselves correctly four general possibilities come to mind, either a defective motor, an incorrect motor installation, a defective linkage, or incorrectly (re)installed wiper arms.
In the situation above it sounds like the motor is not cycling fully (going to it's end position) when it's turned off, so the next time the motor is energized it completes the previous incomplete cycle then starts over correctly (others with more electron experience could probably explain it better).
Point is you won't know for sure what the cause is until you start taking things apart. One quick solution would be to pick up a (good) used or aftermarket wiper motor to have on hand, stuff that in place of the current motor and see how it runs. That way you save the time of pulling everything apart to inspect and don't have to mess with repositioning the wiper arms.
Below is a photo showing the wiper motor unbolted from the firewall but still connected to the wiper linkage.
It shows the correct (down/end) position of the wiper motor lever arm (
Red arrow) pointing at ~9 O'clock (position) while connected to the wiper linkage ball socket pulled slightly out of the access hole (
Green arrows) with the wipers in the full down (correct) position. IIRC.
Here's a link to a previous thread, the hardest part of the process IME was removing the wiper motor from the ball socket of the linkage (if it doesn't just pop out, which it might) so I used a modified tool to help pry the motor ball out of the socket. I discussed clocking the wiper arms in that thread but it appears your problem may not be with the wiper arms if they haven't been messed with.
Thought I'd post up what I found when I pulled the original windshield wiper motor. It sometimes would run slower and was making a sorta growling noise which was more noticeable after I had removed the wiper arms. With the motor apart I found that the magnet on the bottom side of the case had...
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