Narrative design is different to a 'writer' proper. It's a writer specialisation that's about 'higher level' stuff, you could say.
A writer generally speaking, generates content, and that might be on a micro or macro level. They might be given say a character, area or a plot to write, for example. Not always that straightforward, but easy to define as a dev role.
A narrative designer though - they might look at all the game elements that contribute to the story, e.g. gameplay mechanics, art, sound etc, or look at the cast to maintain thematic consistency, emotional impact, tone, mechanics, and possibly acting as a bridge between multiple disciplines to achieve those goals. They usually write a fair bit too. The role can vary a bit depending on the genre, or from studio to studio, depending on the idiosyncracies of how responsibilities are metered out within teams.
