The logout functionality you’ve mentioned here is the default behavior of a plugin. It is logging you out from WordPress but not from your Application/Provider
To allow the plugin to log you out from provider/application account(what you are looking for), you need to make the below setting:
- Go to Sign In Settings tab – > Advanced Settings.
- Add your Application’s logout URL in front of Custom redirect URL after logout.
Note: You will get this logout URL from your Provider/Application support.