A display manager & window manager doesn't even need a protocol
A #Wayland compositor does off courses
Window manager just handles windows and does use a protocol
Only reason I can guess #nixos calls a desktop environment a desktop manager , because of its common variable namespace and pattern for sub name spaces environment, home manger, sessions
While my terminology may not pe be precise is my guessed reasoning correct here
#Linux #GUI
https://vid.bina.me/tools/nixos/breaking-down-the-nixos-gui-setup/