Can a stepper motor flyback diode discharge a coil when connected in serial to the coil and opposite to the current?
The answer to your question really depends on what type of stepping motor you are using.
In this case from the web link you provided they are discussing a unipolar stepping motor where the phase coils are bifilar wound and act very much as a transformer. 

Here is a schematic with the simulation so you can see the waveforms.    







simulate this circuit &ndash; Schematic created using CircuitLab



Notice that the back EMF is clipped by the diode on the OPPOSITE transistor to the one just turned off. So D2 clips the back EMF caused by Q1 turning off and D1 clips the back EMF when Q2 turns off. 

The website is a very poor tutorial since no attempt is made to clarify the differences between stepper motor configurations. The technique for example could not be applied to a 2 phase single winding stepper motor. It uniquely applies to a dual wound (or center tapped) winding configuration driven as a unipolar stepper.