That Black Box looks very much like a cheap clone of a MotoGadget mUnit.
Using momentary switches, you might as well ditch your new, stock harness. The wires from the push buttons are "grounds" and trigger the input side of the Black Box to send power to the output side (+VE or switched 12v) and directly to the component (lights, indicators, tail light, etc). You'll be hacking up the harness a fair bit because it was originally configured to route power through the switch at the handlebar, then onward to the component.
One of your links (Black Box sw diagram) is zero bytes, so can't open it to examine it.
Looks like the k1, k2 wiring is for powering the coils/ignition. Likely routed to a KILL/RUN switch. Normally closed, it would pass +VE through it (RUN), OPEN it severs the power (KILL)