[LATECH]

Yank the alternator off the car, take it down and have it tested, by a counterperson who knows the correct way to test it.
The voltage sounds good, so the regulator is OK.
Most likely problem will be a rectifier gone bad. That is why the volts look OK but the light glows dimly.
I bet it wont pass the current output (amps) test when checked.
I have had alternators that turn on the light when revved up, but werent charging when that happened, that would be the brushes in that instance, yours puts out 13.8 when revved, not going to be the brushes.
When a rectifier goes bad it usually goes open circuit and passes no current, on ocasion a rectifier will short out and pass alternating current. AC is pretty tough on batterys and electrical devices designed to run on DC.
The alternator makes alternating current, the rectifier is necessary to convert it to Direct current.
The rectifier is the aluminum block lookin finned thing in the pic below.