Seems to me that there are no “move” operations in S3, only copy and delete (which mean exactly what they say). So copy still means copy, and if you implement your own “move” you have to make sure it does what you want.
Yep, which is what makes "creates a copy" in this warning seem like "copy and delete". Because just creating a copy and not deleting the original is nothing at all like editing storage class.
