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In the book version, Gandalf confronts Saruman and declares "Your staff is broken," at which point Saruman's staff shatters.

I just re-watched the movie version on DVD, and that scene isn't shown. Instead, Gandalf says Saruman has no more power" or some such, and there is no direct confrontation between the two wizards after Gandalf becomes Gandalf the White.

However, there is a Youtube video I found that is closer to the scene from the book. What version of the movie has that scene in it?

That scene is from the Extended Editions of The Lord of The Rings series.

The success of the theatrical cuts brought about four-disc Extended Editions, with new editing, added special effects and music. The extended cuts of the films and the included special features were spread over two discs, and a limited collector's edition was also released
_{- Wikipedia}

The scene in question:

Scene 4: The Voice of Saruman

This scene was cut from the original film, and was later added into the extended edition from the film, it was one of 4 scenes that were cut entirely from the second half of The Return of the King.

The decision to remove the scene on the cutting room floor was explained by Peter Jackson in a 2003 interview with Ain't It Cool News:

"The trouble is, when we viewed various ROTK cuts over the last few weeks, it feels like the first scenes are wrapping last year's movie, instead of starting the new one. We felt it got Return Of The King off to an uncertain beginning, since Saruman plays no role in the events of ROTK (we don't have the Scouring later, as the book does), yet we dwell in Isengard for quite a long time before our new story kicks off.

"We reluctantly made the decision to save this sequence for the DVD. The choice was made on the basis that most people will assume that Saruman was vanquished by the Helm's Deep events, and Ent attack. We can now crack straight into setting up the narrative tension of ROTK, which features Sauron as the villain."
_{The Guardian - Jackson defends Saruman cuts}

A description of the differences can be found on Movie Censorship, which also discusses it's removal from the film on the cutting room floor.

The scene can be found on youtube, shown below

Christopher Lee's reaction to Jackson cutting his final scene out of the film was rather dramatic, causing a falling out between the two before a rekindling. The clip below shows his thoughts after rekindling his relationship with Peter Jackson and talks about it rather jokingly:

From a 2003 Sky News article, Lee expressed his surprise to his removal and his mood on the premiere:

"Of course I am very shocked, that's all I can say."

"No, what's the point? What's the point of going? None at all."
_{Sky News - Christopher Lee axed from film finale - Wayback Machine}