How to add correct Last update in Postings (the_modified_time)

You could compare post_date to post_modified and only echo your content if they do not match.

// inside a Loop
if ($post->post_date != $post->post_modified) { ?>
  <span style="font-size:85%">Last update <u><time datetime="<?php the_modified_time('d-m-y'); ?>"> <?php the_modified_time('l  j  F, Y'); ?></time></u></span><?php
}

If you want “significant” updates, though, you will need to determine what counts as significant and save your own data. WordPress is going to update that “last modified” value with any update, however trivial. Something like this:

function save_non_trivial_edit_date_wpse_103622($pid) {
  if (
    (defined('DOING_AUTOSAVE') && DOING_AUTOSAVE) 
    || (defined('DOING_AJAX') && DOING_AJAX) 
    || isset($_REQUEST['bulk_edit'])
  ) {
    return;
  } else {
    update_post_meta($pid,'_sig_update_date',time());
  }
}
add_action('save_post', 'save_non_trivial_edit_date_wpse_103622');

Which you’d retrieve with:

// in the Loop
$sig_update = get_post_meta($post->ID,'_sig_update_date',true); ?>
<span style="font-size:85%">Last update <u><time datetime="<?php echo date('d-m-y',$sig_update); ?>"> <?php echo date('d-m-y',$sig_update); ?></time></u></span><?php

Even that is a fairly trivial “significant update” tracker. You can change the update just by saving, even if nothing is changed. You’d have to start comparing revision word counts (or something) to do better. You will have to think that through and work out what counts as “significant”.