Google search for pictures of the atpac and you can see quite a few.
First ones used straps and a huge fiberglass...later changed to a type of plastic housing that surrounded everything. Had a shot box with quick dump that held lead shot and marbles for spaces. That version would have had the bands you have.
Later versions ditched the housing then moved to a close fitting jacket attached to the box that closed with a cumberbun...Later the bladder was streamlined by putting it into a spandex/nylon outer bag that made it a lot smaller. Inflators changed a few times, early ones had the LP attachment near the elbow on a brass fitting...Then a hose ran inside a larger hose to the inflator. This design was notorious for having the inner hose leak a little and constantly fill the bladder very very slowly. Later changed to a standard style inflator.

Doing it right should include some common sense, not just blindly following specs and instructions. .Gary D, AWAP on SB