Could be a faulty voltage regulator.
With the engine running it seems as though you’re measuring the battery voltage and not the regulator output voltage to the battery.
Check the regulator lead output as there should be approx. 13.8-14V. (minimum required to properly charge a 12V battery)
If not with the engine stopped, unplug the voltage regulator and test each of the stator contacts to ground. If the stator isn’t shorted to ground, check for continuity between the stator contacts.
If OK you may have to check the AC voltage to the regulator, with the engine running I think it is >35V