designer, as I assume we design isn't one of Z1s specialties. IMO, the correct action would be to go after the web design company (unless there is a clause in their contract releasing them from liability in this situation) for the $160 loss, not the customer.
Is it the customer's responsibility to notify Z1 of the error? No. But I'm sure many of the more level headed individuals (yourself included) here would have realized there was an error and placed the order over the phone, as opposed to taking advantage of the situation.
That being said, the OP is a toolbag for trying to exploit the error.