What kind of eye protection would I need for this particular spot welding system?
You're right about the arc welding full-face mask - they use very dark glass to accommodate the bright arc. There is no arc here, so in such mask you'll see nothing. Also unlike arc welding (where you spend bit of time aligning, then you lower the mask and then spend considerable time doing the actual welding) here you spend almost all the time on aligning the material but the actual welding takes only tiny fraction of total time spent working (real spot welders use capacitors to do the job in sub-second timeframes. This is where the savings come from). So a vision-limiting full face mask won't work here, because you'll have strong incentives to simply not wear it.

I believe you should use gas-welding goggles, the convertible kind where you can lift dark glass and work with clear glass. Just make sure they are rated for operation with the clear glass alone, because cheap models have only superficial clear plastic that won't protect against larger projectile or UV. And of course, stay clear from "steampunk", "mad scientist", etc decorative goggles because they're intended for nothing more than, well, decoration. Especially now, around Halloween, the market is flooded with such toys.