trouble switching window managers

[junefish]

I'm running Debian 11, based on mrfixit's image.  The current desktop environment is MATE and I'd like to switch (purely personal preference).  I've installed both i3 and cinnamon, but can't figure out how to switch to using them.  There's no button to switch on the login screen, which is what the internet keeps repeatedly telling me to do.

I dug around in /etc/lightdm/lightdm.conf but nothing there seemed to have any effect on window manager; also I borked lightdm somehow so it couldn't start.  I've since purged and reinstalled lightdm.

Here's some further info, please let me know what else I can provide to be helpful:

/etc/lightdm/lightdm.conf

Code:

$cat /etc/X11/default-display-manager
/usr/sbin/lightdm

$ls /usr/share/xsessions

cinnamon.desktop
cinnamon2d.desktop
i3-with-shmlog.desktop
i3.desktop
lightdm-xsession.desktop
mate.desktop

$update-alternatives --list x-window-manager
/usr/bin/i3
/usr/bin/marco

$update-alternatives --list x-session-manager
/usr/bin/cinnamon-session
/usr/bin/mate-session

TIA.