Alternator. First, make sure the alternator frame is grounded properly, and its wires are on tight. If the battery were bad, I don't think the idiot light would come on. That comes from the charging system, & it doesn't know how to detect a bad battery.

Be careful about just disconnecting the battery with the engine running. Voltage spikes aren't too healthy for your ECU.

I'm guessing the voltage regulator. Maybe its brushes are just worn out; 190k miles is pretty good for brush life.