The kind with the single bolt that comes up from the bottom is a more modern design. It squeezes a bunch of parts together, that clamp on the rails from above and below, while also allowing for some tilt adjustment when the bolt is loosened. The benefit is possibly lighter weight, it looks better, and allows for much finer adjustment of the tilt angle. Also, there's less risk of damaging the saddle when turning the central bolt with an allen wrench.

What I'd do is remove that bolt and carefully lift the seat off. You'll see how it goes together, then it will be more self explanatory. Keep an eye on how the parts are oriented and stacked, so you can get it back together the same way. In fact, take some "before" pictures. Even without being able to post them here, they'll serve as a guide in case a part falls on the floor or there's some ambiguity about how to reassemble it.

The new design is more "modern" because it requires cast or forged parts, whereas the old design could be made entirely using metal forming of sheet and tubular material.