The car in question is a 1996 Impreza Outback. I'm guessing that the previous owner had somehow dropped something heavy on the window trim while the window was open and managed to ding up the metal part of the trim. As a result, the rubber scraper is partially pulled away from the glass, allowing water to roll straight down into the door. I'm heading to a salvage yard tomorrow morning and figured I'd try to swipe the same piece of trim off a car there, but I'm not sure how this trim is mounted or how I can remove it without damaging it. I tried searching and couldn't find any information, so I figured I'd ask. See the attached image for a shot of the trim piece.

open the door. on the end of the door at the top of the rubber molding, there is a screw at the end of the trim piece from the inside. peel back the rubber nugget at the top and you will find it.

 

once you remove the screw, you can slide the piece off. hope this helps. this is how it was on an 05 impreza swapping an 02 door.

Thanks for the tip! It turned out that there was a screw toward each end, accessible from the inside. The trim piece was a bit reluctant to slide, but I managed. Looks and works much better now!