One precaution, PLEASE don't remove that battery cable to see if the alternator is charging! The rectifier diodes, and the internal voltage regulator, if it's got one, won't stand much of that. I'm sure you've already done that at least once, or you wouldn't be telling us it works!
The solid-state diodes and transistors, etc., don't like voltage surges, and I've seen spikes of 100 volts or more when a cable is removed while an engine is running. Check the running voltage with a digital voltmeter. It should be from 13.5 to 14.5 or so if the battery is well-charged. Against a low battery, the voltage will be around 13.25 volts or slightly less, and the current output will be 40 amps or more. As the battery builds up, the current drops, and the voltage rises.
Jerry
