I believe I said from the passenger side..I may have made a mistake?
Anyway...The plug that is molded on to the brown wire has the following letters marked on it in this order.
S F L P
S goes to the big pin that you noted on the alternator connection to the regulator. S is also closest to the driver's side fender on my car.
The plug is keyed and should only go on in one direction and has a retainer clip that pops over the nose on the back of the plastic regulator housing that forms the socket on the alternator that the plug pops into. The letters face the front of the car and the retaining clip points to the firewall.
The brown wire from the dash is molded into the plug and goes to L. This is the only terminal in the plug that has metal inserts in it. The others are all just holes.
If you count from the drivers side of the plug when holding it as it plugs in. S (the big pin) is 1, F is 2, L is 3, and P is 4. Therefore L goes to the third pin over from the drivers side which is the same thing as the second pin from the passenger side.
If this is the way yours is connected, then we have a paradox.
The brown wire, when shorted out, makes the volt lite burn brightly in the dash. Therefore the charge initialization circuit is good. If this wire is connected to the correct pin as above, the alternator should charge if it is good. Now, the alternator does not charge. Not even when you applied voltage directly to the brown wire. Yet, when the alternator shop tests it, they say it charges fine.
Something does not compute here.
It seems strange to me that the car starts and then soon dies. If you are charging the battery well enuf, it should run a long time before the electrical system uses enuf power to drain the battery down to that level.
If the wire is going to the correct pin above and it is not charging. I have to wonder if the battery is shorting out inside and killing the alternator output.
You should be able to charge the battery completely up and drive it down to the alternator shop and let them explain why it will not charge.
I had a battery a couple of years ago that tested perfectly..it was a new battery. Suddenly, I would have a dead battery, or it would simply stop running. I would jump the battery and it would run fine again until it stopped..maybe 30 minutes later or two days later....I checked everything...finally took the battery back and they gave me a new one and it has worked with no problems ever since. The battery would even work on a load test. I think it had a bad connection between cells or something that came and went. I have a bad opinion of most batteries.