By default, xscreensaver looks for a service called gdmflexiserver to provide a “New Login” function from within the screensaver, so that a different user than has the screen locked can login.
This does not work with lightdm, which is what MX uses.
This can be made to work with a little extra code in an ~/.Xresources file.
with the following contents:
xscreensaver.newLoginCommand: dm-tool switch-to-greeter
Logout and login, and when your screensaver engages, the New Login Button should work. Note that if you login as the same user as is running the screensaver, you’ll end up putting your password in twice (once for the login, once for the screensaver lock).
Use Ctrl + F7/F8/F9 etc… to switch between sessions.