A simple search on google would suggest the shortcut is ‘Cmd+F1’.

Yes, it works, but it is not precise. The real key combination is ‘Cmd + display_brightness_decment’. This does not make any difference for the internal keyboard, as  F1 acts as display_brightness_decrement by default. However, I am using HHKB, the equivalent for the key is ‘Fn+o’.

Now I can happily switch between these 2 modes using ‘Cmd+Fn+o’. Hurrah!

Thanks to Karabiner-EventViewer during the debugging.