I did something similar, instead of installing the default, I just didn't select Mate, leaving Debian Desktop and System Utilities selected .
I thought Debian Desktop Environment WAS Mate but it wasn't it was GNOME or something that looked just like it and it used a different display manager.
The same thing happened logging in, it went into a loop .
Cindy Sue, I also noticed what you did because I started logging in as root and using MATE, once inside I could use Mate Terminal and use my user account Document files.
I think I started to find the problem but I forget what I did but I was able to log into Mate at reboot after logging into Mate by using the root account and logging out and rebooting.
This joy didn't last long because of problems with Xauthority files, they were causing problems. Didier helped me with this, he said that logging in as root and running X can cause file ownership problems. He advised doing chown to all files to my username,
I had just installed Joe's Window Manager (jwn) and now that I had two window managers, jwm and mate, I had to select one of them.
Now I could log into Mate.
So I purged / uninstalled jwn.
That's when I discovered that if I chose "Mate" instead of "default" I could log in.
So now I only have the problem with all the other consoles not talking!
Time for a few hours of sleep.