Well actually no, battery voltage should be closer to 13 volts, not 12. In fact a battery that has been recently charged will show over 13 volts, called a surface charge. Generator set point charging voltage is higher than an alternator at any given temperature, at least according to the shop manual. This may be due to a generator having more of a "choppy" or averaging waveform? Not sure on why exactly.
The voltage regulator throttles back both voltage and current mechanically via points, they are somewhat finicky and adjustments have to be done with a good understanding of how they work. High resistance in cables and grounds should be remedied before messing with regulator output. Any corrosion will limit current output.