Again, this seems pretty straightforward, but there's one concept that you should be aware of.
Usually it's a good idea when you are editing your site to keep the editor and the live site in two different tabs. That way you can work on your site in one, and view your changes in the other. Using the Home Page tab will take you to your live site, but it will not log you out of the editor. If you are on that community computer, don't use this Home Page button. View your site in that separate tab, and make sure you log out of the editor in the first tab when you're done with your edits.