Based off of the code from another question that you referenced, I’ve added onto it by changing the total number of comment using jQuery:
add_filter( 'the_comments', 'wpse_236055_filter_comments' );
add_action( 'admin_head', 'wpse_236055_comment_notification' );
function wpse_236055_filter_comments( $comments ){ // Display comments related to the author
global $pagenow, $user_ID, $comment_count;
wp_get_current_user();
if ( $pagenow == 'edit-comments.php' && current_user_can( 'author' ) ) {
foreach( $comments as $i => $comment ) {
$the_post = get_post( $comment -> comment_post_ID );
if ( $comment -> user_id != $user_ID && $the_post -> post_author != $user_ID ) {
unset( $comments[$i] );
}
}
}
$comment_count = count( $comments );
// echo '<!-- DEBUG PRINT --> <pre>'; print_r( $comment_count ); echo '</pre>';
return $comments;
}
function wpse_236055_comment_notification( $comments ) { // Only show total count of comment related to the author
global $pagenow, $comment_count;
$site_name="Comments (" . count( $comments ) . ') ‹ ' . get_bloginfo( 'name' ) . ' — WordPress';
if ( !current_user_can( 'administrator' ) ) {
?>
<script type="text/javascript">
jQuery( document ).ready( function( $ ) {
$( '.pending-count' ).html( '<?php echo $comment_count; ?>' );
$( '.comment-count-pending' ).html( '<?php echo $comment_count; ?>' );
} );
</script>
<?php
}
if ( $pagenow == 'edit-comments.php' && !current_user_can( 'administrator' ) ) {
?>
<script type="text/javascript">
jQuery( document ).ready( function( $ ) {
$( document.title="<?php echo $site_name; ?>" );
} );
</script>
<?php
}
}
As a result, the Comments menu item will only show the number of comments related to the author instead of the total number of comments. This has been tested on my end and it works.