The diode doesn't suddenly 'turn on' at 300mV. It begins conducting as soon as it is forward biased. What is important for a detector diode is the difference between the forward current and reverse current at low voltages.
Have a read here: http://www.valveradio.net/radio/high-sensitivity-am-detector.html
Just to add to AalbertHall's explanation, the forward voltage specification applies to a specific current and temperature. This is true of all rectifier types as far as I know and can imagine.
A diode has an exponential relation between forward voltage and current, so will still conduct current at anything above zero volts.
The given "typical" forward voltages of diodes are usually at around a mA of so of current.