Will renaming or updating a product prevent customer notifications being sent?
If you simply update a product, such as renaming it, or updating the description or photos, that will be fine. Back in Stock tracks products in the catalog using the internal Shopify id. This won't change if you simply update a product.
If you were to remove and re-create a variant, or remove and re-create a an entire product, this would generate a new id and we'd need to sync things up at this end.
Note that some tools, such as doing a CSV import to update products, can result in variant be removed and re-recreated on a product. When this happens Shopify assigns a new id to the variant. Back in Stock then can no longer track the updated variant. If this happens just email [email protected] with the name of the products affected and we can sync them up for you.