12.4 volts is a good reading for an "engine-off" measurement at the battery. A lead-acid cell that is fully charged has a specific gravity reading of the acid of 1.250-1.265, depending on the strength of the acid whan the battery was manufactured. A cell in that condition has an open-circuit voltage of 2.1 volts. A fully charged 6-cell battery ("12 volts") will have an open-circuit voltage of 12.6. Yours is close enough to that reading to be considered "full charged".
With the engine running, does the cutout relay close? From the voltage readings you've been quoting, I don't think it would be possible to get the relay closed. The closing voltage should be 12.5 at a minimum, to prevent the battery from discharging through the generator. If the cutout is closing, you definitely need to increase the spring tension on the voltage regulator points to increase the charging voltage.
You will not be able to see the current or voltage regulator points move. they're vibrating at a rate of several cycles per second, and the movement is measured in the thousandths of an inch. You might be able to feel the points vibrate by touching them gently with a fingertip. There's no shock hazard in doing that, BTW.
Jerry
