[tomj76]

Battery light is telling you that the battery isn't getting charged, but instead it is providing power to run the car electrical system. The voltage reading is telling you the same thing (11.4 v when running vs 12.7 v with everything off). My guess is there's a bad contact in the alternator when it's cold, but makes "good" contact after things warm up. Sometimes that comes from a cracked solder joint. Make sure you test the alternator "cold" so that you can tell if that is the cause and not something in the wiring.