Wiring a new light fixture

I have a two switches, side-by side, that control a single light fixture. I am certain this was in the event we wanted to install a fan at some point. After removing the ceiling fixture, I only found four wires: Black, Red, Green, Ground.

I would like to add a new ceiling fixture to be controlled by one of these switches, but am a little unsure of the wiring. After running a new wire from the old box (lightOld) with B, W, G wires to the new box (lightNew) , here is what I think needs to happen in the old box:

Red connects to old light fixture's Black wire
Black connects to lightNew only (old Black to new Black)
White connects to lightOld and lightNew in parallel
Ground connects to lightOld and lightNew in parallel

Is this correct?

Thanks!