All of the responsibilities should go hand in hand with one and another, ideally there should be collaboration on all parties.
A player managing Narrative and Plot may be thinking or working the broad strokes of the game story, or the plot points along the way.
“The player characters are given the chance to negotiate with the uprising scion of a Fey court, in order to prevent a misunderstanding from cascading into conflict. This plays into the larger world of the setting in X specific ways. Details of History and Implications of Possible Futures if players do not intervene are Y.”
A player managing Player Hooks and Spotlight may be more focused on the the more immediate moment of which characters are getting which attention and how much is being presented.
“Character Y has had the spotlight for a good majority of today’s game, do we want Character Z to shine a little here? I remember Character Z had a thing about their mother being related to the Fey courts, this could be a great time to bring that up. Alternatively character X has a strange trinket we never figured out what it does, maybe it relates to this.”