/etc/X11/default-display-manager

At least in Debian, /etc/X11/default-display-manager contains the path to the executable of the display manager, for example /usr/sbin/gdm3.

Compare with /etc/sysconfig/desktop (Red Hat, Fedora) and /etc/sysconfig/displaymanager (OpenSuSe).

grep ExecStart= /etc/systemd/system/display-manager.service