[ray_mcavoy]

That sounds like you have a bad horn relay.

When you push the horn button, it completes the circuit to ground (on the black wire) to energize the horn relay. That should cause the relay contacts to close (connecting the red & green wires) to send power to the horn.

Some relays are louder than others when the contacts close, but yours being faint could be an indication that the contacts aren't fully closing. Or, the contacts might just be bad (burned/pitted) and not making a good electrical connection. In any case, the testing you have done indicates the horn, horn button, and wiring is all okay so replacing the relay should fix the problem.

Edit: By the way, the 3-pin relay used with the AAW Classic Update harness is a common horn relay that was used in many GM & Chrysler vehicles from the 70's up into the 90's. So ordering one for say a 73-87 GM truck should get you the correct part.