Yes, definitely current draw, but current draw is dependent to the resistance to the starter motor pinion gear, ie. engine size, compression, etc.
What happens when the starter motor gets stuck? Current draw increases dramatically, nothing to do with battery voltage.
Actually, a 12 volt battery on a 6v starter will draw slightly MORE current than with a 6 volt battery in the same engine because the starter spins faster with 12 volts (than 6 v) and hence slightly more friction is created in the engine. Do the experiment, use a clamp meter over the battery cable and see the results yourself.
Current = torque
Voltage = spin speed.
Modern engines with 12 volt starter motors have hi torque gearing, less friction, are better balanced and designed to start within the first stroke hence do not need the cable size of yester-years.