BuddyPress.org

Your menu bar seems fine to me without login.

For displaying a shortcut to the logged in members profile in this bar

Create your own function like bp_adminbar_authors_menu for logged in members profile and add an action to bp_adminbar_menus

e.g.

function bp_adminbar_logged_members(){

....

....

}



add_action( 'bp_adminbar_menus', 'bp_adminbar_logged_members', 100 );