taxonomy_edit_form_fields output after term fields

Got it!

It seems this action is fired within a table element.

Removing the _fields part seems to have done the trick as this refers to an action fired later in the page, after the table.

$term_filter_name_edit = $type . '_edit_form';
add_action($term_filter_name_edit, 'box_term', 1000, 1 );
function box_term() {
  echo 'Test output';
}

Reference: https://github.com/WordPress/WordPress/blob/master/wp-admin/edit-tag-form.php