Eh, my choice of wording. Comes from a long experience of role-playing games. Someone who takes up the lead in creating the story and the setting is 'running' the game, because they're putting everything together for their player(S). 
But idk. You picked up on that, but didn't mention any of the other stuff I said about, regarding my experience of telling and playing stories with ponies, so it's up to you really.