First question... Are you working on an old car that is wired with only ONE wire feeding both rear brake lights? If yes, obviously both lights will flash for either left or right.
If this is the case, a second wire needs to be added, so that the brake lights are fed from TWO independent wires.
With that settled, then your question about the black/grey... It DOES NOT go to the hot terminal, but rather to the terminal that is hot when the pedal is depressed. NO wire from the switch goes to either brake light. (The wire that used to be on the switch for the brake lights, now becomes a brake light wire for either L or R, your choice. And, this original wire for the brake lights, needs to be separated, so that it feeds only ONE brake light, your choice. Then, this wire connects to Grey or Black, your choice. The new wire that you added, goes to your 'other' choice, for one of the brake lights.)
Since you didn't ask about any of the other wires, I assume you have those figured out.
(Black wire goes to RR brake light, Grey wire goes to LR brake light. One of those wires will be NEW, if you are dealing with an older car.)