When I got my Lemans last month I stripped it and checked all the electrical connections (I have a Ducati also !!!), the same bullet connector had corroded so much that it fell apart when I touched it. So I imagine you have had the same problem, a corroded connector that due to its increased resistance got hot enough to melt the insulation.I have done a repair with male/female spade connectors and lots of heat shrink, you will need a high wattage soldering iron as the heat bleeds away to much to get the ends of the cables hot enough.My intention is to take the bike to an auto electrician and get a decent weather proof connectors fitted.
When I contacted the previous owner(I bought from a dealer) he said he had noticed the light stay on a couple of times but when the garage checked it was charging fine, so if anybody has this fault then it would be wise to check/clean and grease this connector as it may fail suddenly.
Hope this helps.