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So far we have only talked about two generic parts of diodes when infact there are millions in Si, and Schottky types and many other materials like GaAs and Al In Ga As

But for Si Diodes all the variation in forward voltage, Vf is due to logarithmic current and then linear current x bulk resistance when in saturation. This depends on the area of the chip and process quality controls which is not Gaussian but skewed to the high side of nominal.  Whereas resistance measurements done at much lower currents are more due to leakage effects with high resistance.

But now I am getting into the nitty gritty of diode performance beyond the intent of your exquisite DMM virtual clone.

 

For those who understand, there are many levels of accuracy in model simulations. But on log I vs linear V we see an asymptotic point above the exponential rise in current followed by the linear bulk resistance effects which affect the differences in diodes until you pass a few % of its rated steady current.  The other interesting parameter is than bigger diodes also have bigger capacitance at 0V dc which always reduces with reverse voltage.  This affects recovery time for applications when this matters.  Although I have never had a DMM with a C position during my early R&D career, it would have been useful.