Welllll....
Actually that story about water rationing (forcing the crew to climb the rigging and get the dipper before they could drink fresh water) was one of the few pieces of outright fiction that Nordoff and Hall added to the novel Mutiny on the Bounty. There's no record that it actually happened - and the written record of this story is remarkably thorough. If that story were true, either it would have shown up in the logbook, or it would have come up in one of the courts martial, or one of the numerous private memoirs and journals by the participants would have mentioned it.
Generally speaking, Nordoff and Hall were pretty careful about sticking with history in that book. They changed a couple of names ("Roger Byam" was based closely on the story of Midshipman Peter Heywood) and invented some details to fill in gaps in the documentary sources (e.g., the personal stories about life in Tahiti), but apparently figured out early on that the true story didn't need much embellishment. (The other two volumes in the "Bounty Trilogy," Men Against the Sea and Pitcairn's Island, contain a good bit more outright invention - especially the latter, for obvious reasons.)
The old Charles Laughton/Clark Gable movie actually sticks quite closely to the book. By modern standards the acting is, well, sort of over-the-top, but the narrative actually isn't bad - until the mutiny itself is over. The Hollywood scriptwriters then took leave of history by making Bligh the captain of the Pandora.
The second movie, with Marlon Brando and Trevor Howard, is, in terms of historical accuracy, a mess. (I've always thought Trevor Howard was a great actor, and was perfectly capable of doing a good job of playing Bligh, but the script he was given was pretty awful - and he was about twenty years too old for the part.) The third one, with Anthony Hopkins and Mel Gibson, has a lot to recommend it, but in my opinion it's let down by an extremely disappointing script.
The writer, Robert Bolt, was a great playwright (author of "A Man for All Seasons," among others), but apparently he did no research whatever before tackling the Bounty story. He invented several big dramatic incidents and circumstances (Bligh "replacing Fryer with Christian"; Bligh announcing that he'd decided to go around the Horn on the way to Jamaica; Christian being of a higher social caste than Bligh; Bligh's obsession with circumnavigation) that had nothing to do with reality. And Mr. Bolt apparently made no effort whatever to familiarize himself with the eighteenth-century nautical idiom. (Christian may have said "Get aboard the boat, sir," or "Get on board the boat, sir," or "Get in the boat, sir," but there's no way any professional seaman would have said "Get on the boat sir.") On the other hand, the acting in that version was, at least to my taste, the best of the three. And I liked the fact that the filmmakers turned it into an intimate story of human relationships, rather than a massive "epic," which the first two tried to be.