What I still don't know is how does the sine shape looks like on the output of these converters. I know that this might important too in regarding of the reigniting of the arc after every zero cross.
It is by far not just about the zero cross, but mainly what happens to the voltage shape (so then the rms value), when the arc voltage of a constant voltage clamp characteristic is subtracted from the supply voltage.
With a 100V arc the remaining rms voltage across the filament is normally (with true sinewave supply) about 140V ( =sqrt(arccos(100V/(230V*sqrt(2))/(Pi/2))*((230V*sqrt(2))-100V)/sqrt(2) so in other words a rms value from what remains of a normal 230Vrms sinewave waveform exceeding the 100V of the arc).
E.g. the "modified sinewave" using mostly +325/0/-325/0 pattern keeps across the filament sqrt(1/2)*(325-100)=160Vrms, which is about 10% filament overdrive. And that is quite a lot.
If the inverter generates a +325/-325V rectangle (common for the low power, very cheap inverters intended to supply just the mobile chargers and similar devices), the filament voltage would be 225V, so about 50% overdrive.
With a rectangle of +230V/-230V (to maintain the rms) the filament would be a bit underdriven, but the 230V may not really suffice for a good and reliable ignition.