I-V characteristic of Diodes
I'm not sure that I can totally answer this, but I'll write down what I know.

The equation that you've given is the voltage of a diode, assuming that the diode is not in breakdown. If you apply a high reverse voltage to the diode, all bets are off - it enters breakdown, and your equation doesn't tell us anything about the breakdown region.

So, what's different about these three types of diodes?


Signal and rectifier diodes have the same voltage-current characteristic. As far as I know, the only difference is that rectifier diodes are made a bit tougher for high voltages and currents (think of a rectifier circuit) while signal diodes are meant for lower operating points. They both follow the Shockley equation.
Zener diodes act differently in the breakdown region - while 'regular' diodes go into avalanche breakdown, Zener diodes go into Zener breakdown (go figure), which happens at a specific voltage. Take a look at the reverse voltage portion of a typical Zener diode (and note that the forward voltage part still matches the Shockley equation):