Thank you for your reply and for sharing the screen recording.
However, may I kindly ask if you had a chance to review my previous message? I requested that you show the frontend login page, which is located under the "Pages" main menu on your website.
The issue you're describing is not actually a bug. As mentioned earlier, in order to access or exit the frontend editor, the user must be logged in or out from the frontend—not the Joomla administrator (backend) area.
In your case, you're logging out only from the Joomla admin panel, but still remain logged in on the frontend. That’s why the edit button continues to appear. Once you log out properly from the frontend, the edit options will no longer be visible.
Please try that and let me know if the issue persists.