How to preserve edits to Name or Slug of term when using wp_update_term on save?

So, thanks to cjbj I finally found the correct answer! I needed to use edited_term instead of edit_term. Very subtle difference. edited_term fires after a term has been saved.

// insert stuff into description field of taxonomy
function insert_taxonomy_content( $term_id, $tt_id, $taxonomy ){
    // only insert content on certain taxonomies
    if ( $taxonomy === 'some_custom_taxonomy' ){

         // unhook this function so it doesn't loop infinitely
         remove_action('edited_term', 'insert_taxonomy_content');

         $content = "Some Content";
         $update_args = array(
             'description' => $content,
         );

         // update the post, which calls save_post again
         wp_update_term( $term_id, $taxonomy, $update_args );

         // re-hook this function
         add_action('edited_term', 'insert_taxonomy_content');
     }
 }
add_action('edited_term', 'insert_taxonomy_content', 10, 3);