Rewrite Page with a slug before page name

I’ve not tried external rules, but you can achieve that with an internal rewrite and pagename:

function wpd_page_rewrite(){
    add_rewrite_rule(
        '^anothername/([^/]*)/?',
        'index.php?pagename=$matches[1]',
        'top'
    );
}
add_action( 'init', 'wpd_page_rewrite' );

For parent/child pages you have to set pagename to the parent/child path, page/pagename, or you can also use page_id and use the page’s ID.

EDIT – note that you have to flush rewrite rules after adding new ones. a quick way to do this during testing is to visit the permalinks settings page.