The "capacitor inside the motor" is a thermal fuse to cut all power if it overheats.
Black is one terminal - whether live or neutral is intended I don't know, the fuse
would suggest live, but black as a colour doesn't!
Either grey or yellow is the other terminal and the other winding gets its
phase-shifted current via the big capacitor. If the windings are identical
then it shouldn't matter which of grey or yellow you power. If they are
different it will matter and offhand I don't know how to determine this.
Ideally about the same amp-turns will be present on each of the two phases.
Are you absolutely sure its 240V AC? Just because the capacitor is rated that
high doesn't mean the windings are, it might be 110Vac - without a ratings plate its
rather risky guess work.
And are you sure its 50/60Hz and not some other frequency? (OK, bit unlikely)
You absolutely must add an earth connection to the frame of the motor, given the
dodgy source, and check the isolation of the wiring from the frame, preferably with