do all clipper circuits use zener diods?

no, you can use a signal diode or even a transistor. One well known circuit uses an opamp and a diode.

The diode, for example, is connected to a supply rail, like +12v, so any voltage above 12.6 volts is shorted to the supply and won't go any higher.

The Zener Diode is just one of many ways to clip.

You don't have to use a zener. You could use any type of shunt regulating device.