Correct, the bike will not start if the voltage is too low - the same interlock from the ECU to the starter/headlight relay is common to both low voltage and clutch switch, therefor the symptoms are same.
You may have just got lucky in that it saw sufficient voltage at the time you pressed the button on this occasion.
If it is at 12.58 with engine running, then you have a generator problem (either the stator or the R/R)
But also note that if your resting voltage is only 12.1 then your battery is not in great shape, especially if it is supposed to be 'new'