Strictly speaking, a VOM in resistance mode connected to a non-linear diode is not really a "measurement", but an "indication".
The total circuit is a battery in series with an adjustable resistor and microammeter, connected in series with the diode.
For a real diode, the indication is a function of the details of that circuit.
For a typical DMM, the circuit is simpler to understand: some use a constant-current source in series with the diode, so the display is a measure of the voltage across the diode at that fixed current level.
However, some digital meters measure the ratio of the device voltage to the voltage across an internal resistor, which is not a fixed current through the series circuit.
The original instructions, from when VOMs were still common, merely distinguish a shorted diode from one in reasonable condition, since diodes typically fail short-circuited.