Trackback Count

While this is not totally what you are after, it may be of some help…
https://www.cre8d-design.com/blog/2006/03/27/separating-pingbackstrackbacks-from-comments-in-wordpress-20/

This little snippet will achieve what you need…

<?php /* NUMBERS OF PINGS AND COMMENTS */
  $ping_count = $comment_count = 0;
  foreach ( $comments as $comment )
    get_comment_type() == "comment" ? ++$comment_count : ++$ping_count;
?>