I’m sure someone with a D will come along soon, I can only comment on my S.
When I switch between the modes Green/Mid/Sport there is no option to cycle via the screen; it’s done only on the switch down next to the gearstick.
However I see other options are available via my trip computer/info screen in the Rev counter, but the centre screen itself only previews the mode you have selected.
Green should give you a slower throttle response compared to Mid (default) mode, which I would of assumed all MINIs would have, but again not 100% sure on the Cooper D? Does the throttle feel less responsive when Green appears?
If you have Sport (not sure Cooper D has this) it should sharpen the throttle, and make the steering heavier.