You need to be on the Edmodo website to log out of your account.

If you're already on the Edmodo website, don't worry about this step.
 This will prompt a menu with a bunch of options including the ability to Logout.,

Still appears that you're logged in? Try clearing your web browser's cache and cookies.
Go to Edmodo.com.
Tap the ☰ button at the upper left of the page.
Tap Logout at the bottom of the menu.
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