# To display which display manager is in use
# At least, in a Debian-based system):
cat /etc/X11/default-display-manager
# This file does not exist in Arch

# To change default display manager on a Debian-based system:

# Install an additional display manager; e.g.:
apt install gdm3

# From the Debian Wiki:
# =====================
# Whenever a package containing a display manager is installed
# and configured, it'll try to detect if there's another display
# manager currently installed. If so, it'll open a dialog asking
# which one you'd like to use as a default.
# To change this later, you can reconfigure any display manager
# package that you have installed, and the dialog will reappear.

# Run dpkg-reconfigure on one of the display managers; e.g.:
dpkg-reconfigure lightdm
# or
dpkg-reconfigure gdm3

# You will be shown a screen on which you select which
# of the installed display managers is to be the default.