I think you've adjusted the airgap way too small. You can see there's barely any tube left extending, to stick the plastic hose on. That's not their mistake, it's yours. Those gradations are probably meant to be visible above the top edge of the locking nut, to allow adjustment without taking it completely out.
With the gap correct, the centerline of the lenses will be about the centerline of the airgap. Not with the top electrode extending mostly into the optical path - because then it's blocking a lot of the light from the rear spark unit.
The airgap is supposed to be quite a lot more than you'd expect for 8 to 10 KV. Because the actual breakdown field strength is the main 8KV PLUS the HV trigger pulse, and that's going to be another 10KV or more. Also it's supposed to be acting from a needlepoint at the trigger end.
The way triggered spark gaps work, is an electric field is established between the two major electrodes, which don't have any sharp points. This never breaks down the gap.
The center trigger electrode is supposed to be a needle point, but it's down in the low static field strength area within the end of the ground electrode and the point tip is at the same potential as the surrounding main electrode, so there's little or no ionization on the needle point.
Then a HV pulse is imposed on the trigger needlepoint by the firing circuit. That will be a HV pulse transformer, with a HV cap being discharged through its primary by an SCR. The polarity should be arranged so the leading edge of the spike is the opposite polarity to the main HV (top in this case) sparkgap electrode.
Now the field strength at the needlepoint is much higher, and the air around the tip gets ionized strongly. Those charge carriers are in the main field between the two electrodes, so get swept towards the other electrode. This starts an ionization cascade, and an arc forms across the main airgap.
Ideally, the main arc should form between the big heavy rounded electrodes, not one of them arcing to the needlepoint.
It's hard to see in your video, but it looks to me like the needlepoint has been arc-burned to a flat-ish rough end? And that's why it arcs over to the main ground electrode. It shouldn't do that. Because it's not a sharp point there's not enough field concentration upwards towards the top HV electrode. When you moved the top electrode down a LOT, it got close enough to overcome the effects of the damaged needlepoint tip.
I'd try taking the bottom electrode structure apart completely, and see if you can get the center needlepoint electrode out to reshape or replace it. I'd have expected its height relative to the tip of the surrounding electrode to be adjustable too. Is there a grubscrew or something accessible via that hole at the bottom of the electrode?
Maybe you'll be lucky, and they used something common like a gramophone needle, or a standard thin tungsten TIG welding rod, ground to a point?
But in any case, it's a tiny bit of metal. Can't be too hard to improvise a replacement.