The LSH was not designed for R123 cells since they were not available at the time. Using rechargeable 123 cells is not recommended for the LS series because it could damage the LED.
1) All rev2 models (LS2, LSL, LSH) would drop off rapidly. Rev1 units would drop off more gradually. This is because the rev2 was better regulated.
2) All over discharge protected cells should produce a similiar rapid drop off, unprotected cells would run for longer but the cell would likely be damaged by over discharge.
Pretty much any LiIon Cell with a safety cutoff (low voltage), will shut off the output of the LSH LED with little warning. Those without protection could be overdischarged, damaging the cell.
As the cell approaches ideal depletion, the voltage begans to drop at an accelerated rate for a given current consumption.
A light designed for r123 cells (like the Arc4), will either began to back off the output in advance of the cutoff (thereby extending the time to cutoff), or provide a lower regulated level that allows the cell voltage to temporarily rise above the cutoff (assuming the cutoff doesn't require a recharge reset).
Peter