Hi, When we log-in to WP-Admin, this top admin bar will display.
In the attached image, you have asked to keep the admin menu present on public and admin pages. But, this Menu links is related to Log-in only. Menu Links like Dashboard, Listings, My Profile, Logout... etc. This all links will work once we have login to the website. So, once admin OR public login to the website, we could display this Menu. Without login, there will be no links.
Please let me know your comment.
Thank you.