Indeed that is my mistake SKJ
Thank you for pointing out that distinction! It clears up much of my confusion.

My remaining complaint with Hyatt's site is regardless of point availability if the hotel is bookable with money for the nights you request ; the left side will show the number of points for a standard award night implying availability.
Is in my opinion a major hassle to have to click into each to see if you are able to actually make a points booking. They really should state no available on points just as the hotels with no rooms available show not available.
is a real waste of time ; the APP may provide a better experience? I will give that a try and see.

Regarding the PH Vendome I do see on the 2nd night (10th) it does show a Park Room King and 2 Twin Beds as point options. If I change to 3 (adding a child) those options go away as I assume occupancy is only 2. We could make a 2 twin room work for the 3 of us, but I imagine that is not permitted?

Maybe there is a thread buried somewhere on this forum that lists Hyatt's that allow more than 2 adults as a standard room or points booking at all. Hilton's are similar that some won't work but I suspect has more hotels percentage wise that permit a child.
The Hyatt Regency Paris Étoile looks to be an option at a higher point rate (for a better than standard room) ; club room or suite shows me as available on points.