hook_entity_update is not triggered when checkout is complete

Any order status update triggers an order save, which ultimately calls our entity controller's invoke method where the entity_* hooks are invoked. If you didn't clear the cache before testing your code, the hook wouldn't have been registered. However, you should probably be using hook_commerce_checkout_complete() instead, instead of being dependent on order updates. You never know when an order update may cause an order to pass through the pending status, but the checkout completion hook should only be called once for any given order.