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You want to label the pieces of story by how they are delivered in the game. For example you might have a whole list of item flavortext entries for each item that can be in a player inventory. Then you might have a list of locations, and for each location a list of NPCs there, and for each NPC a list of specific dialogue they say. Then you might have a list of help topics/tutorial tips. And a list of quest text that appears in the player's quest log. This is basically an adaptation of a screenplay style script to an object oriented game system.

If you follow what Sun said you can't go far wrong. Following a screenplay style is a good idea; you just need to remember to include parts that explain the environment the player will be in at that point in time along with a description of any animations going on at the same time. The main difference between an open world game and a RPGs "screenplay" will be the amount of strait dialog you have to include.

As for additional notes to programmers etc. it's largely up to you. Include any information you deem useful/needed. It's important that you're as clear as possible at this point, those reading it aren't doing it for the story but for the raw information.

I have yet to see any standardization between game studies on this, like most things in game design. It's mostly a case of doing/using whatever you want so long as it gets the job done. Some studios may already have a pre-set way of doing things but that is something you would find out if/when you joined them.