How to include non-users in wp-admin redirect?

Question

I added this to my functions.php to restrict non-admin access to wp-admin by redirecting the non-admin to the home page. However, this does not work for users who aren't users (users who are not logged in).

How can I include visitors who are not users in this script?

/**
 * Restrict access to the administration screens.
 *
 * Only administrators will be allowed to access the admin screens,
 * all other users will be automatically redirected to the front of
 * the site instead.
 *
 * We do allow access for Ajax requests though, since these may be
 * initiated from the front end of the site by non-admin users.
 */
function restrict_admin_with_redirect() {
    if ( ! current_user_can( 'manage_options' ) && ( ! wp_doing_ajax() ) ) {
        wp_redirect( site_url() ); 
        exit;
    }
}
add_action( 'admin_init', 'restrict_admin_with_redirect', 1 );

FYI: I created a custom log-in form on a page other than wp-admin so this will not lock out admin

EDIT 2: this is in the functions.php file which may be complicating the issue since when a non-user accesses wp-admin, the server is treating that access as a failed log-in attempt and redirecting the user to the "access-denied" page and not a 404

/**
 * Redirect user on invalid log-in attempts
 */
function login_failed() {
    $login_page  = home_url( '/access-denied' );
    wp_redirect( $login_page . '?login=failed' );
    exit;
}
add_action( 'wp_login_failed', 'login_failed' );

function verify_username_password( $user, $username, $password ) {
    $login_page  = home_url( '/access-denied' );
    if( $username == "" || $password == "" ) {
        wp_redirect( $login_page . "?login=empty" );
        exit;
    }
}
add_filter( 'authenticate', 'verify_username_password', 1, 3);

Answer 1

You can do something like this to redirect everyone that enter to the wp-login.php

Notice that its redirect only when the request is GET so you could still can enter if you post to there. its in the case that you use this wp-login.php with your custom login page.

function login_page_redirect() {
    $redirect_to = "http://google.com"; // url where to redirect
    $page = strtok(basename($_SERVER['REQUEST_URI']), "?");
    if( $page == "wp-login.php" && $_SERVER['REQUEST_METHOD'] == 'GET') {
        wp_redirect($redirect_to);
        exit;
    }
}
add_action('init','login_page_redirect');

Answer 2

Simply add that into if:

if ( 
     (!current_user_can( 'manage_options' ) && !wp_doing_ajax() )   
       || 
     (!is_user_logged_in())
   )