Thanks 1-2 RUN !
I checked carefully, there is nothing "jumped" at all on the old regulator connector. It is just unplugged. The only jumper is on the other alternator (which I don't use anymore) and it is between the "F" on the alternator and the BAT terminal. Is it possible it worked just like that ? I mean it should because I have been running the car like that for more than a year ! Is there a way to "test" my alternator without frying anything ?