What are hard/flat buttons (on appliances) called?

Membrane switch, where one contact of the switch assembly is embedded in a flexible substrate.

Touch pad:

If the touch pads on your microwave oven do not respond but the display lights up, the problem is most likely with the membrane switch. This component, which is more commonly referred to as the touch pad, is actually a series of soft touch electrical switches.

I had originally thought of this as a "touch panel" and Googled that (along with "microwave"). But I then noticed that many of the repair sites referred to it as a "touch pad" (or "touchpad," without the space) instead.

Here, also, is what Merriam-Webster says about touch pad:

a keypad for an electronic device (such as a microwave oven) that consists of a flat surface divided into several differently marked areas which are touched to choose options