The Amazon description for those nuts says...
7/16" wide by about 1" long
so, obviously, they won't work in 9/16" wide T-slots.
Another thing to consider... Real T-nuts are not threaded all the way through. This is to make it impossible for the stud to emerge from the bottom of the nut, touch the bottom of the slot, and begin jacking the T-nut up against the bottom of the ledges. If this happens and enough force is applied the T-slot can be broken. If you ever get T-nuts that are fully threaded, use a chisel to damage the bottom threads so a stud can not pass all the way through.
If real T-nuts are too pricey for you, try buying some carriage bolts with a head diameter greater than 9/16" and grinding them to fit into the T-slots.