The lag swivel is for horizontal wooden beams and pigtail swivels are designed to hook into an eye hook.
Swivels allow for 360° rotation without winding up chair. Some children and adults like to wind up chairs then let go, allowing the chair to spin. This not only damages the rope and chair but is dangerous and the eyebolt can come loose from overhead and fail. Although this might be fun, winding up a hanging chair is not advised.
WITH a chair purchase the price is $18 with shipping included.