I think I would say that the climax of the story is when the police officer (not Jimmy Wells, but the second one who comes to arrest Bob) tells Bob that he is under arrest.
I think I would say that because a story is supposed to have some falling action after the climax, so if the letter is it, then there is no falling action.
In addition, finding out that Bob is under arrest and that the second policeman is not really Jimmy Wells is probably the most tense moment. By contrast, the letter is just explaining what has happened and why. It's more of a "cleaning up" thing that explains what's happened, and that is what falling action is supposed to do.
In the story by O'Henry, a man returns to his hometown to meet a friend he had known years before. The man meets a policeman and engages in a conversation with him. The man reveals how he came to be at the location and how successful he had been in life. He puts down the person that he was going to meet saying basically that he was never too bright.
The climax in the story is when the man opens the letter and learns that his friend was actually the policeman. The policeman had the man arrested by a different man because he could not bring himself to being the one to arrest an old friend. The part in the letter when the policeman reveals his true identity is the climax as you had thought.