How to filter custom taxonomy term name, slug, and description?

The default fields name, slug and description are not term metadata, so you should instead use wp_update_term() to update those fields.

So just replace those three add_term_meta() with:

wp_update_term( $term_id, 'codes', [
    'name'        => $name,
    'slug'        => $slug,
    'description' => $desc,
] );

Additionally, instead of using $_POST, I would use get_term() to get the details of the term that has just been created. So for example,

$term = get_term( $term_id );

// Catch the term info
$name = $term->name;
$slug = $term->slug;
$desc = $term->description;