Is power coming from different parts of the transformer during start up and regular operation? Or is the bulb just putting different loads on the transformer when the bi-metallic strip is in different positions.
The bulb presents different resistance. The transformer supplies a mostly constant current through that resistance, resulting in a variable voltage.
Not only does the bulb have the on/off discharge bypass, but the resistance of the filament increases a bit as it gets hot, and the resistance of the gas decreases as it gets hot. The latter part is why these can't run off of constant voltage. The gas gets hot, and the resistance drops. With constant voltage, that means the current would go up, making the gas get hotter, making the resistance decrease, increasing the current, making the gas get hotter, decreasing the resistance, increasing the current, making the gas get hotter, and so on until something pops. There are other ways around this but current limiting is by far the easiest.
That brings up the note that the open circuit voltage is double the operating voltage of the bulb. The current limiting setup needs some head room, but not that much. The voltage has to be high to arc through the mercury vapor when it's only sort of warm by the filament. The gentle opening of the electrode gap also helps "pull" an arc, but the cold electrodes tend to have an arc cooling effect on short gaps, so the extra voltage helps start it reliably. Also the operating voltage may go up as the electrodes within the bulb wear, or mercury gets absorbed by the electrodes or contaminated with metal from them. Lots of design dependent factors here.
Also worth noting is that the transformer isn't that great at regulating current. The current it can supply is still dependent on the voltage drop across its windings, but this effect is much weaker that the gas temperature/resistance relation and so doesn't have much affect on the stability of the arc.