Why not both? They are both equally true. He said one thing, the world heard another thing. As a historian it isn't our job to pick one over the other. Rather we consider what each tells us about the past/humanity. Why did Armstrong choose to say what he did? How did what people heard affect their view of space exploration/NASA, etc. Historians tend to be more interested in the questions raised by issues like this one.

Professional history is not about the facts. It is analysis of the past based on facts. The facts are the building blocks not the end goal.

He actually said "One small step for man." Listen to the recording. There's no burst of static or audio drop-out, and there's on room in there for an "a." Plus you can tell from his tone that he knew he flubbed his line.

It isn’t the task of historians to produce a canonical narrative of history.

Here's a fascinating first hand account by Joel Shurkin, a reporter who was in the Houston Control Center during the landing. He describes the confusion amongst journalists at the time, and the decision that was made in a moment amongst them to record it without the "a."

"Now we had problem and no time to think about it. Clearly, this was to be one of the most famous quotations in history and we had to get it right. More important at the moment, we had to be consistent. We could not have one news service say one thing, the other two something else, or have the New York Times have one version and the Washington Post another. Forget history, we had to deal with editors.

A small group of us from the wire services and the major papers gathered quickly in the middle of the press room and decided the “a” was missing. It went out that way to all our readers and that is the quote everyone knows.

Later, Armstrong insisted that he knew what he was going to say before he climbed down the ladder and that he said the “a” word. But when NASA cleaned up the recording of the transmission, it was clear we were right and he was wrong. He simply forgot to say it."

He was supposed to say "a man", but he didn't. He flubbed the line. As far as how the quote should appear in print, I don't see what's wrong with "That's one small step for [a] man, one giant leap for mankind."