Doesn't matter, you had the one that matters. As far as I know, the lower the current, the lower the forward voltage of any diode. This could account for a gentle threshold push revealing the polarity safely, same reason a couple of NiMH's can apparently be used on a diode with no resistor! (See Kaz's post about this here). As I said in the reply there, the nominal forward voltage of 2.7V is never reached, so this singular instance is self-limiting. So I guess the question is: is the first hint of light produced at a forward voltage less than the highest safe reverse voltage? If it is, then carefully and slowly raising the voltage (current limited to 50 mA) and periodically switching the LD polarity between incremental raises will safely reveal the polarity of the LD. If it is not, all bets are off unless you like 50/50 odds of destruction.
I just tested a couple of fried Rohm diodes with the diode test, (FRIED, so no worries, the facet is nuked, but electrically they should be viable...), and the forward drops were below 2V at the very low meter test current, so you might get light out of them without exceeding the voltage that would kill them if reversed. I haven't got a load of different diode types so I can't do enough tests to give any idea how widely this can be applied.
Steveo, that Micronta, test the volts on the probes when set to the resistance range (if you have another voltmeter), you might be lucky, it might be around 1.5V. If so, it might be useful.