https://www.ipdb.org/files/2006/Williams_1990_Rollergames_Manual.pdf
On page 94, is the backbox interconnect board schematic

Your bottom row of lights in the backbox is the General Illumination (GI) lamps that runs on ~6VAC. On the same page of the manual shows the GI circuit. So depending on which colored wires go to that GI string, would take you to a possible burnt connection. Start with pulling J6, J7, J9, and J10 to inspect for any burnt pins. Fuse. F1 to F4, could even be blown, depending on the colored wire that goes to that string. Make sure you pull the fuse off the board and test continuity with a meter. Rollergames GI wiring (resized).png

Page 96 shows the wiring a bit better. On the far right, you can see the 'insert board' is the white backbox board door that swings open and has a relay board attached to the backside of the door. If post #2 checks out ok with the fuses and has no burnt pins/connectors, then this relay board will be the next thing to check over.
Rollergames GI wiring page 96 (resized).png

Are you referring to the controlled lamps for the W-I-L-L-I-A-M-S letters?

Page 97 shows the interconnect signals better going to the relay board. Yes, pin 6 is correct.Rollergames Backbox interconnect signals (resized).png
Rollergames solenoid table (resized).png

I do not think your problem lies with that brn/red wire. That is the signal from the CPU to turn off/on both backbox GI strings at the same time. If it was not turning off/on, all the backbox GI lamps would be out. If you unplug the 2 pin connector from the backbox GI relay, the GI lamps should stay on all the time.

If you already checked over everything in post #2, you can also have burnt pins/wiring (of the other 4 wires) at the relay board in the backbox or even bad solder joints on any of the parts on the relay board.

You could swap the backbox GI relay with the playfield GI relay (mounted on the underside of the playfield, likely close to the back left corner if I recall) to rule it out. They are the same board. Or just resolder every part on the back of the backbox GI relay, as there could be broken solder joints.

Time to change a connector/pins. Glad it you found a reason the GI string was out.

If you had moved the wire from pin 6 to 11, that now ties the 6VAC GI to flasher 6C, which is a different voltage, 18VDC, if I recall. The two voltages will clash and I suspect a blown fuse if you are lucky.

Well, it is quite possible the manual is wrong. Need to hear from someone with a Rollergames. I sold mine more than 2 years ago.

Yep. I do miss mine but you can't own them all.

Here's a photo for you. It looks like the manual is correct and for an early production Rollergames.

IMG_2632.JPG

You're mis-reading the numbers. I watched the relevant part. The single wire in the video is connected to pin #6 NOT #11 as you've written. The pin numbering starts at the right and increases to the left.