ah huh!
yep. a scooter. road legal.
best bet is a multimeter, a small flat screwdriver to gently bend connectors, and some contact cleaner...
but. ok. you turn it off at the KEY, or the on/off switch on the bars? the main key should kill everything. the blackwhite is earthed, the power is diconnected from battery. im not familiar with your exact scooter, but they are fairly generic... the only relay is usually for the electric starter. maybe more fancy for the dashboard with all the modern gizmos.
by simply unplugging the key on most pitbikes and scooters, they will run, albeit with no lights etc.
you say you have power, and have to pull the battery, so the first place to look is at the main switch. it sounds like its not switching off. how strange. seriously.
if you can pop the lock(often requires dismantling half the front end as it is a security thing), get it on the table, then do a google on the actual part, you should find basic terminal wiring, then be able to test with a multimeter.
if it has a kicker or something, try starting it with no main switch in the loop... or pull plug, check for spark by spinning engine with a socket on the cranknut.
could simply be a crushed wire like the 240ac live mains wire was squashed between case and chassis on my new plasma cutter!