Whatever story you have heard, it’s not true. “Ich bin ein Berliner” means exactly what JFK wanted it to say. The fact that Berliner is a jelly donut in other parts of Germany (not in Berlin, mind you, here it’s a “Pfannkuchen” or pancake) is completely irrelevant to the point he made.
It’s the equivalent of saying in English “I am a Frankfurter”, if you were showing support for the town in Kentucky. It may sound odd to others not familiar with the place, but the people there would appreciate the sentiment.
Some people have suggested Kennedy should have said “Ich bin Berliner” (I am from Berlin) rather than “Ich bin ein Berliner” (I am a Berliner). But let’s say in the aftermath of the 9/11 attacks, German Chancellor Gerhard Schröder had made a speech in Manhattan in which he said “I am a New Yorker.” Do you think anybody listening would think he was saying that he was a magazine? Everyone understood that the primary meaning of a Berliner is a person from Berlin not a jelly donut and it was that meaning Kennedy was using. Kennedy was not stupid - he had fluent German speakers vet the line.
“I am Danish.”
IIRC, this was beaten to death in a previous thread or several. The consensus from the real german speakers IIRC was that the article is optional in that German sentence and does not have the same nuanced difference as in English?
NOTE: To clarify
I am a Danish - here Danish is a noun. In english, as a noun, Danish is short for “a Danish pastry”. the noun to describe a native is “Dane”.
I am Danish - here Danish is an adjective describing I.
Removing the article means the word is an adjective not an noun.
The equivalent would be “I am a Dane”.
Whereas with Berliner, the same word is the noun “someone from Berlin” and noun “a jelly donut”.
Apparently this is true in (some parts of) Germany too.
So the question revolves around the article “ein”. Vas is los?
About the article “ein”: had Kennedy said “Ich bin Berliner”, he would have said that he is literally from Berlin (or lived there a long time). The article is what makes the metaphor work in the first place. I can’t quite put my finger on it, but there is a difference. It’s not really grammatical, but to a native speaker it just feels more appropriate given the context. I, for instance, would not say “Ich bin ein Berliner” because I actually am one. Plus, in the context of a speech and its catchphrase, if you will, the original just flows much better.
The major source of the “jelly doughnut” interpretations comes from the inspired standup comedy of Eddie Izzard. He’s definitely not above a comedic outright lie for laughs, and if his audience isn’t sophisticated enough to get that it’s all a joke, then the joke’s on them. Oh, and Engelbert Humperdinck really is dead, really, no really, google it, yeah, that’s the ticket.