Let me either make use of the description area inside a custom taxonomy I am building but relabel it to bio, or remove description altogether and make the brand new area by myself.

Any idea what I have to do either of those?

How about this jQuery hack? Put it in functions.php of the theme or produce a simple wordpress plugin:

function rename_category_description() { 
    global $current_screen;
    if ( $current_screen->id == 'edit-category' ) { ?>
        <script type="text/javascript">
        jQuery('document').ready(function() {
<?php }
add_action('admin_head', 'rename_category_description');