'One small step for a man': Did we mishear Armstrong's first words on Moon?

Armstrong insisted that he actually said, 'That's one small step for a man.' In fact, in the official transcript, Nasa transcribes the quote as 'that's one small step for (a) man'

On July 20, 1969, an estimated 650 million people watched in suspense as Neil Armstrong descended a ladder towards the surface of the Moon.

As he took his first steps, he uttered words that would be written into history books for generations to come: “That’s one small step for man. One giant leap for mankind.”

Or at least that’s how the media reported his words.

But Armstrong insisted that he actually said, “That’s one small step for a man.” In fact, in the official transcript of the Moon landing mission, NASA transcribes the quote