Is there a definitive way to determine the positive and negative leads of a laser diode?
The only sure and safe way is the data sheet, but that sounds out of the question.

First off, be careful.  Laser diodes, even without collimation optics can generate enough light to damage your eyes, and the ones you find in a lot of electronics are either infra-red or very deep red that is barely visible.  This means they can be generating damaging light without you realizing it.  In my opinion, the most dangerous lasers are the ones in the 700-800 nm region, which you may be able to see as a very faint red, which can be dangerously misleading.

That said, how many pins does it have?  Many laser diodes have a built-in photo diode as well for measuring and feeding back the output power. Sometimes they have 4 pins, but more often 3, with a common cathode or common anode, or with the cathode of one connected to the anode of the other.

You can work out the pinout using standard diode testing techniques, but you have to be careful.  Laser diodes are very delicate.  The reverse breakdown voltage is usually only a few volts, while excessive forward current can instantly damage the output facet, turning your nice laser into a crummy LED.  A standard multi-meter diode tester can destroy a laser diode.  So you want to make sure your test circuit limits both the voltage and current to safe values, and be careful of ESD.  It is common in laboratory use for laser diode drivers/mounts to use a relay to short the pins together until everything is connected and powered up to make sure that no turn-on transient drives too much current through the diode.

In a 3 pin laser, the photodiode will be a standard silicon diode with a forward voltage drop of around 0.7 volts.  The laser diode will have a forward voltage drop that depends on wavelength, and will generally be around 1.5-2 volts for red and near IR lasers, up to 3-4 volts for blue lasers.