In most system I played with, if your character survive the larp, they gain experience points that then they can use to buy more skills that are available in the rules. For example if i play an alchemist with an alchemy skill lvl 1, i could be able to roleplay the creation of basic potions (defined by the rules, with ingredients provided by the organisation - usually false flowers, stuff like that), and if my character survived the larp, i could maybe buy the lvl 2 offering me better recipes to know.
Of course it is not universal depending the type of larp you play, the format, etc.
So, I think your definition of larp is correct if incomplete : there is a more-or-less rough script (the story), lot of improv (the interpretation) and a set of rules to complete all of that and make it work