A diode is "off" for 0 volts - and, of course, when it is reversely biased.
In case of forward biasing, the diode starts to conduct for voltages > 0 V.
However, the voltage-current relation is exponential and the current for - let`s say - 0.3 volts is remarkably smaller than for 0.6 volts.
It is an oversimplified view to think that the diodes are "off" for voltages < 0.6 Volts. That`s simply wrong!
In particular, for smooth amplitude limiting in a WIEN oscillator the voltage-current relation for voltages < 0.6 volts is important.
Otherwise (on(off at 0.6V) , we would have a hart limiting effect and we wouldn`t need the diodes at all.
However, the real behaviour of the diodes allows a smooth increase of the feedback factor for rising output amplitudes.
As a result, the total harmonic distortion of the output signal will be rather low (depending also on the ration of both feedback resistors (100k/22k)
(Please note that the oscillator was named after Max WIEN - not "WEIN".