This thin wire only carries enough current to illuminate the warning light bulb, 2.2watt so does not need to be thick.
This live feed through bulb then excites field windings in alternator thus starting the charge process, once started it is not needed as it also then outputs positive, two positives do not make a light work and so your light goes out!!
If it discharging , at any time then then it will be the DIODES(small electrical one way valves) in the alternator contained in the RECTIFIER of it.
Odd that it should do this only when running though not impossible, it's(rectifier) primary function is to stop AC current(-) from reaching battery but has the added bonus of preventing back flow of power from battery to alternator.
So check fan belt is tight as if not could be using more power for car than you are putting back when running, then if OK replace rectifier/alternator.
A rectifier is something easily replaced(solder) around £15.
tabetha