If the battery light is on while the car is running, firstly, the alternator won't be charging. Not necessarily because its knackered however.

The alternator uses the battery light to commence charging, if the battery light stays on for whatever reason it won't charge.

So, the alternator either isn't charging, OR, the wire has shorted to earth.

Best way to test, is unplug the thin blue wire and see if the battery light comes on, if it does, you have a short, if not, the alternator isn't charging