I think the google link may be out of date. They stopped providing hotels for stopovers in September 2009 from my understanding. It may be possible that they continue to offer a special service to First Class...but their website booking engine has a very clear statement that Etihad no longer provide this service.
I think you may have been misinformed by the customer service representative you spoke to. There is extensive discussion about Etihad's decision to drop this service in the Flyertalk forums, so I am pretty certain that this is no longer offered.
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