I waited for a snowless day, with temperature above freezing to venture taking a look at the problem. Water was accumulating under the front passenger seat, under the rug, threatening to turn the car into a mobile swamp. To find a leak, your only option is to spray every corner of the car with water and patiently observe and listen where the dripping comes from. It turned out that the water was coming from the trunk.
Under the rug in the trunk, both the spare wheel and the jack compartments were completely underwater. Driving the car was enough to jiggle the water under the back seat. Water would accumulate under the passerger front seat, due to gravity.
Step 1: Drying and Testing
The procedure is to spray the, beforehand closed (!!), trunk with the water from the pitcher. Get someone in the car, the back seat pulled down, with a flashlight to listen and watch where the water comes from (I managed to both spraying and spying). The picture shows what I saw: the water was coming from the backlight!
Step 2: The Demon of Duct Tape
The first thing that came to my mind was, of course, to use duct tape. It sort of worked, but not so well... Indeed, it is difficult to make a perfect seal with duct tape on a surface with acute angles, so I went for something a little more sophisticated: tub and tile adhesive caulk! I found that stuff (that looks like a big toothpaste tube) at my local hardware store for less than 5 bucks. I rubbed the opening with the the caulk and smoothed it with a popsicle stick.
You're supposed to let the caulk dry from a couple of hours to a day (depending on what you bought). Because my car is parked on an open air parking lot, I couldn't leave things like that. I therefore reassembled the backlight, and to insure that things would stay dry, I duct taped it to prevent the water from getting onto the caulk.
Step 3: It's All Good