I finally removed the extension, closed the browser, then reinstalled the extension. It started to work properly.
I noticed this time on the Web-Auth tab, there is a check box that says “Remember Me”. If you check this box, it will never ask you for the key again, ever! The only way I know to uncheck it is by reinstalling the extension.
If you explicitly logout, it should clear this setting.
Actually, my suggestion above will also work (deauthorize sessions from the web vault).
You can think of the 2FA “remember me” option as a declaration that your device is to be trusted — i.e., having possession of the device is the second factor.
Logging in to your Bitwarden vault serves a different purpose. It establishes that both authentication factors are presented, which authorizes your device to store a local copy of the encrypted vault, and to send vault changes to the cloud database. Logging out purges the local vault cache and revokes the authorization to modify the cloud data. However, it does not automatically revoke the status of your device as a trusted second factor.
The solution is simple — avoid the “Remember Me” option.