I am not so good at reading engine serial numbers, but look to the cylinder bank left side sitting on the bike. If it says 1052cm its a 86-88 motor. If it says 1127cm it is either a Katana (black) or GSXR (kinda brownish) engine. Many have been repainted by now so hard to tell. Serial number guys might be able to chime in.
Seeing that mess of wiring through one of your pictures I'd by far suggest you contiue to look there as long as the bike seems to be turning healthy off the starter. No reason to waste time if it is working good. But....
If you want to eliminate my concern, which again looking at that mess of wires I'd imagine the issue is within there. Left side lower engine cover just under those cc numbers I told you to check remove it there is a 17mm bolt holding the starter clutch that can come loose. If it is, best to clean bolt and crank threads, add loctite and with bike in 5th and foot on rear brake try and torque it down 110lb/ft or the best you can do.
I'll add that bike being a 1992 with an 1100 in it, appears to be flatslides, and a dirty but polished frame hiding beneath, it is worthy of finding a new harness. Try and locate one and everything should plug and play right in and offer long term reliability. You've got something nice on your hands there worthy of a little investment.