The coil voltage is below zeners clamp but with zeners attached to circuit node zeners conduct without clamping voltage being reached.

How can the clamp be adjusted to conduct only when coil voltage exceeds clamp voltage?

Unclamped:

Clamped:

 

A zener diode in the forward direction conducts like a normal diode - about 0.7V.
You need to connect another three zener diodes in series with the existing three, but the other way round.

See the double diode clamp circuit here.

 

Wouldn't a cathode grounded 1kv R6 diode be better?

It depends on the intention of the clamp. This way only the positive peak will be clamped while the negative peak will be unaffected.
The way I suggested will clip both peaks symmetrically.