"Quick question... when the vehicle is actually running, is the power being used coming from the battery or the alternator? Just curious."

It's supposed to come from the alternator. The battery and alt are hooked up in parallel so either will generally do the job if the other has quit - BUT normal voltage when running is between 13-14V and the battery will not provide much more than 12V. Maybe 12.5 or so when fully charged and not being given much of a load? So you see the 13-14V only occurs when the alternator is running. It comes from the alternator.

I suggest you run a lead from the battery's + terminal into the cabin and hook it to your voltmeter. You can run the tester's negative ground wire to the metal tab at the rearward end of the auto transmission shifter housing. That way you can drive and see what the voltage is doing. That might start to tell you something more than just watching lights go dim.
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Sven: '89 245 NA, 951 ECU, expanded air dam, forward belly pan reaches oem belly pan, airbox heater upgraded, E-fan, 205/65-15 at 50 psi, IPD sways, no a/c-p/s belt, E-Codes, amber front corner reflectors, aero front face, quad horns, tach, small clock.