FWIW the beginning of my travails with ST1300 ground issues started right there with a crispy connection like yours of the green (ground) wire to the left headlight bulb 3 wire connector. If it were me, and it was, I would purchase a 3 pin headlight socket to replace the OEM one and take the green wire out of the 3 wire connector directly to a solid frame ground near the front of the bike. IIRC there is hole in the frame next to the head stock that is drilled and tapped for a 6x10mm bolt that can be used for a short run from the headlight connector to ground. Alternatively you could use one of the 8 mm bolts that fix the LH coil to the frame. Now is the time to examine the right hand headlight connector, the yellow mini buss above the RH headlight that gathers multiple green grounds together, and the white 24 pin connector found on the left side of the bike above the valve cover. It tucks into a holder built into the left middle cowl. Look for any suspect connections of green ground wires at those places and tend to them now.
