I, too, have a rattle in that seat when the seat is all the way back, and upright. But not when it's down and not much when it's all the way forward.
From the videos, it sort of sounds like the adjustment handle that releases the seat to slide forward or back. Mine is very loose when the seat is upright and back, and is a bit noisy when I jiggle it with my hand. Though the noise also sounds a bit like two things rubbing against each other.
For my noise, I originally thought it was the FWD itself, but now I think it's the seat itself jiggling and making a noise where it contacts the floor. If I strike the seat with the palm of my hand (not too hard), I can get it to approximate the noise. You might try the same thing, since it's hard to figure it out from the driver's seat.