What lights do you use personally?
Before I took Cavern/Intro, I had no interest in caves at all. They were just wet rocks, and that's stupid. I took the class to help my trim, buoyancy, and sidemount skill (I took it through Edd's shop). So, I bought the cheapest can light I could find that was of good quality. It was a Dive Rite 10W HID that had really unhealthy batteries, and I got it off of eBay for like $150. Great quality light, but replacing the batteries was going to cost a fortune (like >$200)....but I had some batteries from another hobby that I rigged up to work. It works surprisingly well for what it is.
I've dove with lights by Light Monkey, Salvo, Halcyon, Dive Rite, and UnderWater Light Dude. I now own an UnderWater Light Dude light.....and if you try to take it from me it'll end VERY poorly for you

. I'll put it against any 21W HID any day of the week. It seems like the only company that can claim to compete with Light Monkey's level of service is UWLD. Fans of UWLD say Bobby treats them better, LM fans say the opposite. I think UWLD lights are unparalleled in efficiency, output, technology, and reliability. A good friend bought a LM 9W LED canister. He likes it. He also owns two UWLD canister lights now
As for the options you were given by PM:
Nickel vs Lithium (NiMH/NiCd vs LiPo/LiFe/LiIon) - This is battery chemistry. This one is simple. Lithium packs are MUCH better. They're lighter, smaller, pack a better punch, and they have a discharge curve. However, they're more expensive and can be more dangerous if mistreated.
HID vs LED - This one is REALLY the tough one. HID is often focusable and is hailed for the tightest of beams.....beam tightness is great for cutting through muck and signalling. However, LED is much more efficient (lumens per watt) and much more robust. HID bulbs have the bulb and ballast to replace, which are often expensive. Also, many designs are fairly fragile. With HID, you also have to be fairly conservative about how you use it.....the bulbs go out after a certain number of "strikes" (times you turn them on), but that number can be reduced drastically by mistreating. If you let them overheat or if you turn them on and then back off before the bulb is allowed to get to temperature, you can blow the bulb or shorten its life. LED is practically bulletproof. Some people trade off that ruggedness for beam tightness and compensate by being more gentle/careful with the light head. Also, LED light heads are typically smaller than HID light heads.....and some HID light heads are huge. That doesn't bother some, it does bother others. All stuff to consider in the HID vs LED argument.