How to know if a tag has been added within a taxonomy?

Could you be more specific on the “when a new term/tag has been added” part? Are you doing it yourself, as in you call the backend through admin-ajax.php and update the front-end accordingly? Including some code or a link to jsFiddle wouldn’t hurt.

There’s a create_term action you can hook into in the back-end. Just put this into functions.php or your plugin code:

function your_create_fn( $term_id, $tt_id, $taxonomy ){
    // do your thing here
}
add_action( 'create_term', 'your_create_fn', 10, 3 );

Alternatively, if you’re doing it yourself then you obviously control the entire process, so you know exactly when you’ve successfully added a new term and can return something to the JS function that indicates you need to clear those values.