Examples of Four Types of Narrative:
Evocative Narratives: Nintendo’s Super Smash Bros.‘ core gameplay is focused on knocking other characters out of the ring, but as a crossover of famous characters from many bestselling Nintendo games, this game evokes the characterization (and fanbases) of these franchises. For example, you can play as Yoshi (debut in Super Mario World) or Isabelle (debut in Animal Crossing), with power-ups and moves that are based on their in-game characterization (e.g. Yoshi is excellent at jumping, Isabelle whacks people with an umbrella).
Enacting Narratives: Stardew Valley is an open-ended game with possible overarching objectives for the player to achieve, such as acquiring varied items for bundles and money to purchase new devices. The player creates their own avatar to meet localized quests (such as fetching a vegetable for a townsperson) which can strengthen friendships and unlock recipes, and meet other goals and festivals as part of creating their ideal farm. They can also choose to romance a bachelor(ette), name a pet and horse, and level-up as a better farmer!
Embedded Narratives: Inkle’s Frankenstein involves reading Mary Shelley’s original fiction work with the framework of 19th century visuals, and the user is able to interact with the “letters” in physical format and flip through pages and other objects. The game follows a branching path system, in which you can follow the main “real” storyline (the embedded narrative) or explore different routes and what-ifs for the story.
Emergent Narratives: The Sims 4 is an amazing sandbox world that is deliberately designed to generate as much conflict as possible, especially with the emotions system where Sims are liable to carry out dramatic decisions, such as slapping another character or forgetting to eat as they cry on the doorstep. As Jenkins mentioned, its structure is open-ended so players can decide what they want their characters to pursue, even as certain decisions are streamlined, like sink interactions limited to washing dishes and hands, or limiting household objects to include a cupcake machine but not interactive pillows.