“I am a New Yorker” and took him to mean “I am a New Yorker magazine.”
So why have smug people been making this claim for the past 20 years?
The earliest reference anyone’s been able to come up with is the 1983 spy novel Berlin Game. A fictional character claims that Kennedy said he was a doughnut. In reviewing the novel, The New York Times treated it as a reference to an amusing fact, rather than a reference to a completely made-up fact, and to this day, you can’t say “Ich bin ein Berliner” in a room full of educated people without having them shout something about a jelly doughnut at you.