Another facet of this that we need to acknowledge is that there is another side to this idea of "story"-- the story that the players experience and the story that the individual characters experience.
For many players/DMs-- stereotypically I'd say of the old-school variety-- that feel that the story is what the player experiences over the course of a campaign, with less concern for the individual characters within that campaign. This is why a number of DMs couldn't care less and in fact fully expect innumerable PCs dying over the course of the game, and that is perfectly acceptable and part of the game. Because it's not about an individual character's story and place within the narrative of the campaign... it's about the player who experiences the narrative of the campaign (and all the trials and tribulations the player through trying to get characters through it.)
And lots of different types of campaigns lean closer or further from these two things. If you run a game like Candlekeep Mysteries where you do each module one at a time, that is definitely a "campaign", it just leans more towards "player story" rather than "character story" because the story of these modules is not going to put any narrative load onto the individual characters or be character-specific. Rather, it will end up as a narrative the player will experience. After all, the player could choose to play a different PC for every single module and it wouldn't affect anything-- there is nothing narratively to connect one adventure to another for individual characters. Whereas an adventure path like Curse of Strahd leans more towards "character story" rather than "player story", because most encounters are designed to forward a narrative for the characters that are involved-- Escape Barovia! Escaping this hellscape is something the individual characters want and need, but the players themselves probably couldn't care less (other than escaping being the "win condition" of the campaign, if "winning" is that important to the player.) Whether the players adventure in "Barovia" or adventure in "the Sword Coast"... that doesn't really matter. Playing the game is playing the game for the players, regardless of how/where the DM sets it.
Adventure paths and sandboxes are the same way... one leans more towards character wants and needs, the other toward player wants and needs. None of them are just that, obviously... the best games will give both the player and the player's characters things to hang their hat on when all is said and done... but the focus does tend towards one more often than the other. And depending on who you are as a player and what you find important will influence the types of games you would prefer to run and play.
For many players/DMs-- stereotypically I'd say of the old-school variety-- that feel that the story is what the player experiences over the course of a campaign, with less concern for the individual characters within that campaign. This is why a number of DMs couldn't care less and in fact fully expect innumerable PCs dying over the course of the game, and that is perfectly acceptable and part of the game. Because it's not about an individual character's story and place within the narrative of the campaign... it's about the player who experiences the narrative of the campaign (and all the trials and tribulations the player through trying to get characters through it.)
And lots of different types of campaigns lean closer or further from these two things. If you run a game like Candlekeep Mysteries where you do each module one at a time, that is definitely a "campaign", it just leans more towards "player story" rather than "character story" because the story of these modules is not going to put any narrative load onto the individual characters or be character-specific. Rather, it will end up as a narrative the player will experience. After all, the player could choose to play a different PC for every single module and it wouldn't affect anything-- there is nothing narratively to connect one adventure to another for individual characters. Whereas an adventure path like Curse of Strahd leans more towards "character story" rather than "player story", because most encounters are designed to forward a narrative for the characters that are involved-- Escape Barovia! Escaping this hellscape is something the individual characters want and need, but the players themselves probably couldn't care less (other than escaping being the "win condition" of the campaign, if "winning" is that important to the player.) Whether the players adventure in "Barovia" or adventure in "the Sword Coast"... that doesn't really matter. Playing the game is playing the game for the players, regardless of how/where the DM sets it.
Adventure paths and sandboxes are the same way... one leans more towards character wants and needs, the other toward player wants and needs. None of them are just that, obviously... the best games will give both the player and the player's characters things to hang their hat on when all is said and done... but the focus does tend towards one more often than the other. And depending on who you are as a player and what you find important will influence the types of games you would prefer to run and play.