I would guess that the threads are bottoming before the end of the body is making contact with either the head or the tailcap (or both). I've not had this happen to me, but I've seen enough variation in L2P threads that some heads won't thread properly onto some bodies, so it's certainly plausible.
Look at the light with the L2P/L1P body combination and be sure there is a slight gap between the head and body and tail and body at the junctions. If there's not a gap, the parts could be bottoming out before making electrical contact. The head should screw onto the end with the wider space next to the oring.
Try unscrewing the ring inside the tailcap 1/2 to a full turn to see if you gain a connection. A pair of circlip pliers works well for this.
If you have a multimeter, you can use the continuity check to see which end is not making contact. Thread only the head on, and check continuity between the PCB and the tail end of the body, and then thread only the tail on and check continuity between the tailcap contact and the head end of the body. This will show you which end is giving you issues.
