MOVs have a smooth, but nonlinear, curve from not conducting at low voltage to high conduction current at higher voltages. They do not "trigger" like a neon light. And the voltage across the MOV does not suddenly decrease, like it would in a neon light (you probably didn't say it did). "Clamping" is a widely used term, including the wiki article on MOVs. (Gas discharge tubes are like a neon light, and do trigger.)
One of the parameters for a MOV is MCOV (maximum continuous operating voltage) which is the voltage at which the current is 1 mA. The increase in current is smooth (but very non-linear) above the MCOV, just like it was smooth (and non-linear) below the MCOV.
(When the MCOV for a MOV decreases 10% it is the defined end of life for a MOV - referred to in the wiki article.)
A MOV is very much like back-to-back Zener diodes, but does not clamp as sharply. But MOVs have huge current capacity in a small package.
The clamp voltage that is usually cited is the UL let through voltage (UL calls it something a little different). This is the voltage at a specified test surge current. If the surge current goes up, the let through voltage will be higher (in a non-linear way).