Immersion is trumped by safety.
You need light. Period. You must have a well lit area or the combat is dangerous. Make sure everyone is aware of the "safety" call. There's nothing you can do but have big lights -- projectors are great apart for the poor sod who is in front of it. You can use filters (red is good, black lights are good) to make sure that you get more of a feel of night time. Filters do work on individual lights (for example colour glass on mocked gas lamps).
Anything is fine as long as you are still safe. You must have enough lights that it is safe to move about. Flash lights are good, so are mock up oil lamps. Do not allow naked flames (torches, oil lamps, petrol lights). Candles are fine provided that they are set on stable surfaces. Again, no fighting around candles.
Charcters' own lights
In a fantasy with magic LRP, I have seen a wizards staff (not combat safe) with batteries and a cluster of LEDs on the top mimicking a light spell. Easy to do, looks great, but you cannot fight with it. It would be easy to do "orbs of light" in the same way. Otherwise, we have jokes of mecha-bushi -- wind up torch, samurai character, call of "wait, I need power" followed by howling laughter...
Note that it is not a mag light, it is a wand of light project…
In non-fantasy or more modern LRP, then picking something appropriate is not hard.
Accidents do happen.
I did experience a LARP where bad light did cause a collision between a (well hidden) BBQ and a leg resulting in an open fracture and arterial spray. Thankfully, there was a trained nurse on site or it might have ended in someone's death. So, safety does trump immersion for me and all those I LRP with…