Here is the update.
The whining noise is coming from the voltage regulator. When this noise happens, the battery meter is jumping up and down.
I tested all connections and grounds. All checked out. I even added a ground from the negative battery terminal to the body of the voltage regulator. Still no change. I bypassed the voltage regulator and confirmed that the alternator is pushing 14 volts, and the battery reads 14 volts.
I replaced the voltage regulator again. The one that I took off had signs of heat on the back side. Put the new one on. Start up the truck and the volt meter shows 14 volts at the battery. No generator light; no noise. Turn it off, and then back on. Generator light comes on for a few seconds. At the same time, the regulator whines, and the volt meter drops to 12.5. Then, on its own, the generator light goes off, whining stops, and volt meter goes back up to 14.
I continue to be at a loss, and do not know if I should be concerned, or just leave it alone as the battery is charging.