A quote by Marc Davis in Didier Ghez's book Walt's People Volume 10, “Walt knew that we were not telling stories. You know, he and I discussed it many times. He said very definitely ‘You can’t tell a story in this medium’” prompted me to write down some existing difficulties I've found with defining story in the theme park sense, and how it is variously used to describe different elements, even when used by a single company like Walt Disney Imagineering. This is certainly an area I feel I will expand on in some detail in the future, and I have already written a piece critiquing the common view that Pirates of the Caribbean has no story. The quote was posted by Edde Sotto in his excellent thread on the WDWMagic message boards.
I think the topic of Story in theme park attractions is difficult because the word can be used in so many different ways. I'd essentially break it down into three facets:
BACKSTORY - The fictional explanation of how an attraction came to be. Imagineering seems to use this definition when it discusses the 'story' behind Prince Charming Regal Carrousel (he had the contraption built to practice jousting) or attractions like California Screamin' which don't have anything actually happen on them narratively.
STORY PREMISE - I'd think this is the 'old school' definition of story within Imagineering. Essentially its a set-up: "what if you explored a haunted house", "what if you sailed with pirates in the Caribbean". No explicit narrative is communicated but an environment, theme and atmosphere is created, peppered with events, moments and vignettes that populate the world. A specific narrative is open to translation by the audience.
STORYLINE - This is the modern definition of story used by Imagineering, which seems to have come directly from Eisner's background in film. It's a movie style plot of explicitly connected building moments. Nothing is really open to interpretation because everything is communicated and linked cinematically. Every audience member receives the same experience. Generally these attractions can be slotted into a story structure, most popularly "...And Something Goes Wrong".
(I would certainly say that it's a sliding scale between the three. How many events needed to be added to / removed from Space Mountain for it to switch between backstory and premise for example).
The best example of two and three I think is the Haunted Mansion (DL/WDW) vs. Phantom Manor (DLP). In the first, we are the main character; it is about our experience as we tour this haunted house stumbling across events (the seance, the party, the hitchhiking ghosts) that impact on us, even remotely. In the Phantom Manor however, it's no longer our story but the story of Melanie, the bride. As we move through the house we are given pieces of the puzzle that reveal her story. We're not peeking at 'A' ghostly ball (which we can claim as OUR ghostly ball), it is HER ghostly ball. This switch from internal/implicit story to external/explicit story was particularly damned by Marc Davis.
Now I'm one of many who thinks the third has vastly dominated the second in the past two decades, but I do think the third has a place in the theme park even though so many fans berate it. Because of that I wouldn't jump on Walt's quote as proof of anything other than how open 'story' is to interpretation.
Maybe Walt meant story as in film and literature; the introduction of characters, their internal and external struggles, their dialoguing, their adventure and so on, which of course is vastly unsuited to the theme park. We know that Marc detested the third style, but I'm not convinced we can use 'Walt's authority' to write it off. Star Tours, Muppets and Splash are all examples of effective attractions that use the third definition of story, Splash particularly - we are ENTIRELY external to the story, passive observers following it along, the emotions we feel (other than the non-diegetic anticipation of the big drop) being essentially mimetic. Really - why on earth is this log we're floating in disconnectedly following the story of Brer Rabbit; do we feel part of the story other than sharing the plunge Brer Rabbit endures to escape; how much better is this than just watching the film*? But it's still popular.
* Assuming the DVD was released!
Follow-Up Note: Since writing this I've been quite seriously looking into story within theme parks and my ideas have grown quite significantly. I hope to post a (much larger) update to this sometime in the future.