The only thing disconnecting the battery cable when running tells you is that your alternator is producing enough current to power the ignition. An MSD 6a requires 1 amp per 1000 RPM (on a V8). So at 6000 RPM you'd need 6 amps.

Headlights alone can draw 40 amps. Add the wipers, defroster and hit a turn signal and your draw is 70+ amps. That's why the heater motor slows down on those icy mornings when we have lights, heater, heated seats etc on and then turn on the rear window defogger. The battery compensates - but only slightly as it is stored energy.

A lot of new vehicles come with 100 amp plus to accommodate accessories and the killer stereo systems.

Your "warm" alternator indicates a shorted diode..