If you haven’t larped then this distinction is difficult to understand.
Larp is designed around the personal experience of a player so is very individual. There is not a definitive record of “what happened.” There is a running joke in the nordic scene that (Finnish) people have “immersion closets” which they go into alone to do some intense larping.
It is also very much what you put it rather than being given to you. This guardian review presents an interesting example of an immersive theatre/larp culture clash.
Another angle to discuss this might be historiography. If you are doing political history then your understanding of a particular period might be the narrative of a powerful figure. This is theatre. If instead you’re a social historian then the idea of “one story” doesn’t really make sense. There might be broad themes or movements but there are thousands of individual stories. This is larp.