Can not change desktop environments

I am trying to check out different desktop environments, so I installed Debian 12.4 and included several of the desktop environments during the installation. Immediately after installing, I am using the gnome desktop as shown by the following command:

$ echo $DESKTOP_SESSIONgnome

Everything I find on the Internet tells me that I can change my desktop environment with the following command:

sudo update-alternatives --config x-session-manager

So, I ran that and selected the Cinnamon desktop, and then restarted the system. After I login, nothing about the desktop environment looks any different than before. I don't think the command had any effect at all. The $DESKTOP_SESSION environment variable still says gnome, but update-alternatives thinks I am using the Cinnamon desktop.

How would I know if the desktop environment really changed or not? I doesn't look like it did. The only different is that the environment variable and the update-alternatives now think I am using different desktops.

How can I actually know which desktop environment is in use, and how can I change from one to another reliably?