I have purchased cheap "sine wave" inverters that were actually square wave.  the odd thing is that don's Radio shack inverters are modified sine wave (square wave) but seem to have a unique waveform compared to the other ones.  Radio shack ones like he used are cheap ($15-$30 used on ebay).

search for "ne-5p neon" instead of a variac.

Don Smith used what he called impedance conversion where he hooked his (HV) output RF coil directly to the very low impedance twelve volt battery circuit.
This was completely irrationally insane from the normal electronic design practice where the high voltage would have much higher impedance. Then he needed the inverter
to step the maintained battery 12VDC back up to 120VAC 60hz to run neon sign transformer. The variable autotransformer 120VAC Variac sat between the inverter
and the neon transformer, acting like a throttle, to control how much his output coil was excited.

In a normal design the high voltage RF would have to be rectified and tediously stepped down to regulated 12VDC at relatively high current so as to maintain efficiency.
Where does the high current DC need to run the inverter come from?

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Why doesn't he use mains power? He wants you to be impressed when the system keeps running without being plugged in. ie. proof of OU in the system!