Actually not in this case... the Diamond guaranteed is applied when the hotel is fully booked and they will have to walk someone else with a valid booking. That justifies the higher price.
Apparently the hotel has some (wholesale) customers that they have sold a number of rooms to. And those rooms are still available on the 3rd party websites.

Since Hyatt has no control over this capacity they are still fully booked even if there are rooms available elsewhere. Therefore the advice to book with the 3rd party website seems valid (or pay the inflated guaranteed rate).