These camera use a curtain (focal plane) type shutter. It has two curtains, one that travels up and one that travels down.
Incorrect. Both curtains travel in the same direction. Prior to pressing the shutter button they're both at the top. One moves down to open the shutter to the film/sensor. Then, after the correct delay, the other moves down to close the shutter.
When you cock the camera (manually or automatically) for the next shot, both curtains are brought back to their starting positions.
While reading further, bear in mind that an electronic flash is VERY brief.
At high shutter speeds (brief opening times), the miles per hour speed of the curtains becomes relevant. The second curtain begins to close before the first curtain is fully open. Thus, you never have an instant when the shutter is 100% open. Instead, you have a traveling slit. If the flash fires, you illuminate only that slit, which begins when the flash is triggered (when the first curtain has reached its full open position).
The limitation was worse with older cameras with curtains that moved sideways (because they must travel further).
Leaf shutters do not have this limitation because there is always an instant when the shutter is 100% open.
The primary benefit to a FP shutter is with NON-flash photos. Even with the traveling slit, the entire frame is exposed uniformly. A leaf shutter opens from the center and then closes back toward the center. Consequently, the portion of the frame closest to the center is always exposed more than the portion near the edges.
Years ago, we had "FP flashbulbs," which were long-burning bulbs. The FP sync fired the flash and then opened the shutter and closed the shutter while the bulb continued to burn. SOME electronic flashes (e.g., Nikon SB-600 and SB-800 with the correct camera body) simulate long-burning bulbs with a series of a large quantity of low-power flashes.