Question with get_the_term_list

You could use get_terms() and loop through it – using the term description as the “short name” and slug as the specific class ( or you could use an array of colors of your choice ). Let’s look at a quick example:

<?php
    $terms = wp_get_post_terms( $post_id, 'category' );

    if( ! empty( $terms ) ) : ?>

        <?php foreach( $terms as $term ) : ?>

            <div class="<?php echo $term->slug; ?>"><?php echo $term->description; ?></div>

        <?php endforeach; ?>

<?php
    endif;
?>

If you wanted to make a list out of it or link from it you have full control over the HTML – let me know if you have questions about the markup.