Photodiode operating modes - confusion about wording
1, and 2 define the photovoltaic mode to be the mode where a short circuit is enforced across the PD terminals (by the OPAMP).

The picture you linked for 2 is describing what the transimpedance amplifier circuit in the picture is doing (holding the voltage across the diode at zero), which causes the photodiode to be in photovoltaic mode.  It is not defining anything.  The picture in 4 is not of a transimpedance amplifier, and so the circuit does something else.

1.)How would you guys interpret &quot;zero-bias&quot;? Is it a short circuit or an open circuit? I assume this is a key for further discussion and understanding.

Zero bias means that you are not applying a reverse voltage to the diode.  It does not imply either short circuit or open circuit operation, just that there is no externally applied voltage.  Both of the pictures you presented do not apply an external voltage across the diode, so they're both zero bias.

2.) What is the reason that the PV mod as described in 4 is not used in practise? 6 just also mentions that fact but does not explain why

That quote does explain why.  The signal you get is the log of the optical signal and requires very sensitive circuitry to read out.  That makes it hard to use.

3.) PC mode seems to represent 3rd quadrant, what about PV mode? If used open loop I would say 1st quadrant, but 7 suggest it is the 4th quadrant?

That is a plot of current vs energy (not voltage).  In that picture, quadrant 1 is an LED (energy going in, light coming out).  Quadrant 4 is the reverse, light coming in, energy going out).