That "button" is the barrel of a narrow hinge. A "domino" keeps the parts align when the chair is deployed. It's very clever how it was made to be a folding chair. There's a brass pivot hinge that allows the arm/seat pair to rotate and a brass support bracket that rides under the arm to support it when the chair is deployed. Here is one of the few photos I've ever found of a collapsed chair that reveals some of those details.
