Was JFK a Doughnut?

On June 26th 1963 U.S. President John F. Kennedy spoke to the people of West Berlin in a statement of solidarity.  The result has become a much derided phrase:

"Ich bin ein Berliner"

 What JFK actually intended was to declare "I am a citizen of Berlin".  Ein Berliner, however, is a popular variety of doughnut.  The implication of this being that he should have said:

"Ich bin Berliner"

Nationalities are not treated as "countable" in German.  You can't individually be many citizens of Berlin - unless you have split personality disorder - so this makes sense.  The limit is set by the personal pronoun rather than the noun's article, so the article is removed and the noun treated more like an adjective.

I've never had any reason to question this...until recently.

Listening to a BBC Radio 4 podcast I was casually informed by a translation expert that this little joke at the expense of the late President Kennedy was actually utter nonsense.  "Ich bin ein Berliner" means exactly what was intended.  JFK was not claiming, in a moment of madness, to be a doughnut - metaphorical or otherwise.

Now, I'm not one to argue with the BBC.  So I'm calling for debate.  Can anyone shed light on this claim?  All I see is doughnuts, but my ears and eyes are wide open.