LSI = leveling strength index
BSI = battle strength index
BSI i know is calculated by (attack + defense) / level
Basically, LSI applies to beginner players that put there stat points in energy and stamina to keep a constant leveling speed, while BSI applies to have low energy and stamina and put more stat points in attack or defense.
Ofcourse once you reach a certain level/stat points these names are not so relevant nowdays.
Nowdays, LSI is used for a player that has 2000 energy / 1000 stamina, no matter of how much attack and defense he has (specially if he is high level with a lot of those). And other names appeared, like Half LSI (1000 en, 500 stamina) or Double LSI, Triple LSI, Infinite LSI.