coding a WordPress AJAX Form using PHP to check if User is Logged Out and Show error

A response is needed from php side after sending an AJAX call.

// ... rest of the code

if ( is_user_logged_in() ) {
    wp_send_json_success();
} else {
    wp_send_json_error( null, 403 );
}

// ... rest of the code

And then in your JS code add a new check for jqXHR.status == 403 error.

Documentations:

wp_send_json_success

wp_send_json_error