truly branching narrative

The more I try different games that claim to have branching narrative, the more I realize that the term doesn't give much information to the player.

In general, what this means is that there is a system in which the player can choose options and those options allow them to go down one branch or another.

The problem I see is that in most games that promise branching narrative, the branches are practically cosmetic.

At most, they result in one ending or another, but the story beats remain the same.

I believe there are 2 types of branching narrative that are not cosmetic:

1. Where the choices significantly modify the story, changing not only details but also which story beats are part of the story.
2. A linear story without variations, but where each branch provides information that is not available in the other branches.

Can you think of any other type of branching that works well and doesn't leave the feeling of "lack of agency"?