Yeah, that's a good point, but he also says this: "So that's who I said it was. When Aldrin came back, he told me no, first thing Armstrong did was pass the camera back to me."
It's still hard for me to believe they were passing the camera back and forth. That, with all the other inaccuracies, lead me to believe it's fake. Why someone would purposefully fake that I don't know...
Due to a glitch in the video it's not clear which astronaut is which leading up to the photograph, but if you watch them afterward when they talk through what they're doing, it's clear that Aldrin was the one posing next to the flag.
This is actually rather important. But I guess there is a high chance he misremembered? Hm. When they sent out a correction this should be able to track down?
I wouldn't rely on that for any real information.
EDIT: Not to mention the fact that the suit says "E Aldrin" on it. I have a hard time believing they swapped suits, too.