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 l........