Here are some musings on how one could approach the question. This is hardly a recommendation, and even less an explanation of the "right" way to do things. It's simply a process that seems to be effective.
The key advantages are that it is iterative, that it starts at a high level at the same time that the designers are mulling over what the game will look like, and it tries to ensure that design and story develop together rather than separately or serially.
First Document: Pitch
This is really a marketing/business oriented text with the setting, a few main characters, the basic story arc, and how it fits in the series/IP/brand. It's a 2-3 page overview that has 0 gameplay elements.
Second Document: Synopsis
Here the story gets broken down into macro chunks: Campaigns/settings/levels, major characters/NPCs, and maybe major missions/arcs within the overall plot. Gameplay may be included, but it's more along the lines of ideas, suggestions, possible new features, etc. Up until this point the designers have been working on their pre-production issues, so the process shouldn't be slowing them down.
Third document: Scenario
Now we start getting to the point where we have to think about gameplay. Here the story is translated into the game chunks that the player sees -- campaigns, levels, objectives, maps. Because at this point we are creating the quests and NPCs that will drive the player's actions, integrating the story and the gameplay is a necessity.
However, since the developers have contributed to the documents created so far, and since in practical terms we are already exchanging ideas of how to make certain plotlines/quests work, we don't run into the "not invented here" syndrome.
From here on in the classic writing elements finally come into play: Writing dialogue, doing the in-game texts, refining the characters and the story details. Without the upstream part, however, the story will feel pasted on to the gameplay, and the player risks seeing scenes and hearing dialogue that have nothing to do with the actions he is taking.
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