What is the name for a flap that increases the wing's area but not its camber?
Useless flap.

It is always the increase in camber that increases the lift. The reason to increase the chord is that if the curvature was increased too much, the flow would separate and the wing would stall, so the flaps move aft to be able to move down more while still maintain moderate curvature.

The Fowler flaps do move a bit down even in the first step, and the slats move forward and down with (or even before) them, so the camber increases even in the first step.