"On the SD replacement, green and bare go to ground, black is hot and goes to the volume pot."
So where do I solder the ground to? And does the black get soldered to the back of the bridge volume pot or to one of the lugs on it?
It looks like your original pickup has a single insulated inner core, which is its hot, and a braided outer sheath, which is its ground, and which is soldered to a pot casing.
The new hot (black) goes where the old hot went - to the lug on the bridge volume pot. It's hard to tell from your picture but usually the hot input goes to the middle lug.
One of the outer lugs is connected to the selector switch. It's the red or brown wire in your picture, I think. Leave that one alone.
The other outer lug must be grounded to a pot casing. If you break that existing connection, just bend the lug over until you can re-solder it to the casing or add a short piece of wire to reconnect it.
Generally speaking all "ground" connections go to pot casings. With your new pickup that would be green. The bare wire also ALWAYS goes to ground.
If, as I said, you find the sound odd, thin, in the middle position, your new pickup is
out of phase with the old one. That is to say the hills on the new pickup signal are occuring in the valleys of the old pickup signal. Ideally you want the hills and valleys occurring at the same time, or
in phase. You can either leave it as it is for something like the Peter Green (early, very early, Fleetwood Mac) effect, or swap the black and green wires over. Leave the red and white connected to each other and ALWAYS ground the bare wire.