I agree.I wouldn't think that 12 volts would be enough to strike the arc. Perhaps the "12 VDC inverter" shown can produce a high voltage to start the arc before supplying the 12 volts running voltage. But the capacitor in series method is fairly obvious.
The cap applies 170 volts peak to strike the arc and then limits the current once the gas ionizes. The inverter does the same. It applies a voltage high enough to strike and then limits the current.
The lamp, itself, sets the final voltage drop in both the cap and inverter versions.