In ~/.xsession-errors, I get the message that the SESSION_MANAGER environment variable not defined. In the user manual, there is no indication of what that environment variable should be set to, not even a mention of it.

It should be set by the desktop session eg gnome-session. Applications linked with libsm use the variable to work out how to interact with the manager so it can ask them to save and restore their state. If there's no session manager you may get that warning, but otherwise it doesn't matter unless you want your desktop and applications to always start up in the same state.