Does saving trigger my other plugins?

Saves go through WooCommerce's product, order, coupon and comment APIs, which fire WooCommerce's own hooks, woocommerce_update_product included, on every save. A few older extensions only listen to the classic product edit screen (save_post, woocommerce_process_product_meta_*); tick the box for it under Settings, in Tell other plugins about a save, if such a plugin needs to see the change. The editor never fires a hook twice for one save: when WordPress has already fired save_post (a renamed product, say), the editor does not fire it again.