Looks like your P60L is operating correctly, or close to it... Both of your G10s are operating VERY incorrectly if those readings are true, but since your P60L module is reading at least close to what it should be, I think we can say for sure that both of your G10s are WAY overdriven...
I'm not surprised you are talking about a heat problem, what does surprise me, is that the modules still work at all, they should have self-destructed the LED long ago, unless those readings are incorrect.
Keep in mind that the CR123s you are testing with appear to be running a little on the low side (2.9V open circuit indicates that they are probably fairly discharged, no surprising since your G10s are eating them for lunch it would seem).. You can't take the open circuit voltage and multiply by the running current to figure out power consumption, because the cells will sag under a load, measuring voltage while the circuit is running would be required to get a true total power reading, this can be tricky to say the least
. So more realistically, your P60LED module is probably running closer to ~3W give or take.. (which is where it should be)... The 0.8A draw of the P60L, provided this is an accurate reading, tells us that the input voltage has dropped enough for it to start drawing more current to offset the voltage loss. Under normal circumstances I think it's more like 500-600mA on a pair of CR123s (this from memory, I could be a bit off here)... Since it appears that your G10s are just operating in direct drive (as I had suspected before, even though they aren't supposed to be) we know that the power supply us sagging to the Vf of the emitter at that current, which is probably around 4V, the variation from module to module in this case is a result of the variation in LED Vf, so your actual power consumption is more like 15W and 9W respectively. Regardless, this type of heat and load is all around not safe. The fact that your 3.0V cells won't light up the higher power consuming module is no surprise, as the voltage would immediately sag low enough to kick in the over-discharge protection.
So...
I have some bad news:
* You should not use those 3.0V rechargeable cells anymore. If they have been used to drive even the lower powered module, they have undoubtedly been over-taxed far too much. Even a few discharges at those rates (~4C) would be enough to make them a liability. You should dispose of them properly.
* You also can't use your new 3.7V cells when they come in with either module, as that would be very unsafe as well. More than likely they would trip their current limit and not work on either module anyways, but either way, don't do it.
* You have experienced illumination from severely over-driven LEDs, it's likely that when you get correctly working modules that you may not be all that impressed anymore... I have a hunch about how they are getting that "10K lux" figure now... On a pair of CR123s in the 2+ amp range, an R2 would produce about 10K lux at 1 meter in a 26mm reflector. properly performing R2 modules in this size class will perform ~5-7.5K lux depending on drive level.
*The company that sold you these modules has put you and your home at risk of fire/explosion and hydrofluoric acid poisoning. Discharging CR123s at nearly 4 amps, discharging lithium cobalt's at 4C.... Lots of danger here, you are lucky you haven't had an incident. I would be writing a really nasty email to them, and informing ebay of the dangerous products they are selling.
A few comments:
*When you get your 3.7V cells, it's very important that you do NOT use the charger that you used for your 3.0V cells to charge them. It will over-charge the 3.7V cells to dangerous levels. (the 3.0V cells you have are actually 3.7V cells with voltage "bucking" circuits/diodes in them, charging them requires higher than the normal 4.2V cutoff to over-come the voltage bucking circuit in reverse, so the charger actually terminates at 4.4-4.5V, very dangerous for other cells).
* you may be able to use your G10 modules in a configuration where you have a single 3.7V cell (like an 18650 or the like) driving the module. You would want to take a current measurement on a fresh cell and make sure it isn't any higher than about 1.2A to make sure it can run like this safely, but this is the only way that I would suggest using these modules without some sort of fix...
*If you're handy with a soldering iron, you might consider gutting the LED modules you have there and installing a proper driver. Probably would be best to install new LEDs as well, as they are very likely nearly toast at this point anyways...
-Eric