Couldn't seem to get the video to play. Are you using the welder as a power supply to heat a cutting wire? When I was building RC sailplanes with foam core wings we used a lamp dimmer fed from an isolation transformer to heat a wire stretched on a bow. We followed a Formica template pinned to the foam.
Just a second. The wire feed is set on 0, and voltage turned all the way down, gas off. Pull trigger and turn voltage knob up until it's glowing and go.
The video worked for me. Very interesting. My MIG will still push wire at the 0 setting,
and I do not have infinite current control. But, my Miller buzz-box does. I should
try that, maybe with some NiChrome wire I have.
I used to be in the ICF biz, and we used all sorts of cutting methods for EPS foam: hot-wire
cutters, chainsaws, routers, recip-saws, even heating elements.
I needed to cut foam boards to make sign lettering.
I used nichrome wire* that I scavenged from an old drier element.
Made up what looked like a wooden 'C' so that I could string the wire vertically between 2 porcelain stand offs.
I had left extra coils of the element strung out along the top of the upper arm as I was not sure of what length of element would provide the exact heat setting.
Power was from a 12 volt battery using clips for contacts. Heat adjustment was moving the clip along the coiled element until the appropriate setting was found.
Cutting the foam with heat seals the pores for a nice smooth finish ready to take paint.
I primed with water base primer and top coat with glossy enamel.
After 8 years of exposure those letters still looked just great.
While most folks don't know it, many large sign lettering is foam cut and glued to building facades.
Some city bylaws demand 3-D lettering and this is one easy cheap way of complying.
* nichrome as it withstands heat without melting while copper generally will when it gets hot enough.