Richard Connell's "The Most Dangerous Game" is one of those short stories which rather breaks the conventional plot structure of a short story, and there could certainly be some disagreement about what event qualifies as the climax/crisis/turning point of the story.
The inciting action is when the action begins, and the rising action is everything that happens after that which leads to the climax of the story. After the climax is falling action, followed by the resolution or denouement. If we were to follow this traditional form, the climax would have to be when General Zaroff announces that he intends to hunt Rainsford or perhaps when the actual hunt begins. That reading would allow for plenty of falling action and at least some resolution. This story does not really have a long resolution, though, so a more useful reading is as follows.
The inciting action is either when Rainsford drops his pipe and falls off the boat or when he lands on the island. The rising action goes on for a very long time, until it peaks with Rainsford jumping off the cliff to the rocks and water below and then reappearing in Zaroff's bedroom. Very little story is left after that, but it is enough to resolve the key issues: Rainsford lives and Zaroff dies.
The climax of a story is
the moment in a play, novel, short story, or narrative poem at which the crisis comes to its point of greatest intensity and is resolved. It is also the peak of emotional response from a reader or spectator, and it usually represents the turning point in the action.
Given this definition, a case could be made that Rainsford jumping off the cliff is the climax; the case could also be made that the moment of "greatest intensity" is when Rainsford appears from behind the curtain in Zaroff's room. It is certainly a great surprise to us--and an even greater surprise to Zaroff. The resolution and denouement are only a line or two in either case, but that is enough to ensure that we know who wins the battle.
Either reading seems defensible and depends on what the reader views as the "peak of emotional response." The direction of the story turns when Rainsford escapes from Zaroff, but everything really turns when Rainsford wins the game and Zaroff becomes the hunted.
The general sucked in his breath and smiled. "I congratulate you," he said. You have won the game."
For me, the climax of this story is when Rainsford appears from behind the curtain and wins the game.