Notice: Undefined index: url in purge_logging() (line 202 of purge.inc)

This is the only instance of the text 'url' in the module, so it certainly looks like a mistake.

Based on the comment, I suspect that $purge_request_result['url'] should be $purge_request_result['proxy_url'].

      // Collect all proxy hosts that are blocking the url requests
      case 405:
        $purge_error_log[] = $purge_request_result['purge_url'] . ' on ' . $purge_request_result['proxy_url'] . ' Forbidden';
        $purge_blocking[] = parse_url($purge_request_result['url'], PHP_URL_HOST);
        break;