So opening the visor mirror sliding door did make a difference to the rear wiper: the wiper stopped moving altogether. Closing the sliding door made the wiper start moving again thoug not as one would expect (it moved super slow).

So I think you guys nailed it... I am going to order the ignition switch. Gonna go for the URO brand as it is less than half the price on the BMW original part.

Thank you!

__________________
2001 BMW X5 3.0i, Automatic, Gasoline