I don't think this is accurate. You can check out and check in, but that doesn't give you an additional stay. In fact, you could even be forced to change rooms and it would still count as a single stay.

A tricky case might be where you check out and leave the property with all of your belongings bu then return later that day for a newly made reservation. An example might be that you go the the airport, discover that your flight is cancelled, and want to return to the Hyatt for an additional night.