The display manager is more so what controls the main graphical login process after Debian/Mint/Ubuntu boot and controls the graphical login sequence. Once you login, you are then usually passed to an Xorg based Window manager like XFCE, Mate, Ubuntu etc...
Popular display managers are mdm, gdm, lightdm etc... and they all basically do the same thing with a different interface/style and some feature differences.
In Mint for example the normal default display manager is lightdm and it is defined by this file:
Here are the contents of "default-display-manager":
What really makes a different is your window manager, whether that is XFCE, Mate, Ubuntu etc.. and is controlled in this file:
If you wanted to use XFCE instead, you would change "user-session=xfce" and then restart the window manager (eg. systemctl restart lightdm).
You can also choose the Window Manager before logging in by clicking a button near the login area, which will show you the available Window managers (so for example if you wanted to login this time using XFCE, you could select that).