@MzDoomsayer: It's not referring to a specific location in the world, but rather the martial arts world, the kind with super human strength through training in techniques or breathing or whatever.
Like if a wizard from Harry Potter world introduce themselves as being from 'that side' sort of thing.
Usually they are aware of the damage they can cause in modern world so they try to avoid society and form their own little society, that's Murim.
Example: https://thebreaker.fandom.com/wiki/Murim
So rather than Moyong 'family', it'd be more appropriate to use 'Clan' I think.
Basically after Serenia's self-introduction, Valerie realized Serenia isn't just a tomboy heroine of a romance novel, she's a very much trained warrior and could very well be that 'tiger on fire' her teacher prophesied about so she left in a panic.