echo CHILD / SUB CATEGORIES of the CURRENT CATEGORY only

Your code loops through all top level categories, then shows all child categories of each one, so I’m not surprised it’s showing all child cats!

If you want to show the child terms of the current category (i.e. on a category archive?), use:

$current_cat_id = get_queried_object_id();
$kids = get_terms([
    'taxonomy' => get_queried_object()->taxonomy,
    'parent'   => $current_cat_id,
]);

And as a complete answer:

<?php

$terms = get_terms([
    'taxonomy' => get_queried_object()->taxonomy,
    'parent'   => get_queried_object_id(),
]);

echo '<div style="height: 200px; text-transform: uppercase; border:1px solid #666666; padding:10px; overflow-y: scroll;">
<div class="breaker-small">Refine Search</div>';
foreach ( $terms as $term) {
    echo '<p class="filters"><a href="' . get_term_link( $term ) . '">' . $term->name . '</a></p>';  
}
echo '</div>
<br />';

?>