John Ericson, which is what Khan was named in earlier drafts of Space Seed. And that is indeed what the actors spoke while filming STID, as can be heard whenever his name is spoken in deleted scenes. But then in post-production Orci worried Trek fans would hear the name John Ericson and figure out he was Khan and so the name was changed to John Harrison, and indeed all the actors had to rerecord their dialogue and the new lines saying Harrison had to be dubbed into the movie.
Then everyone figured out the character was Khan anyway due to how similar Harrison sounds to Ericson.
I liked How It Should Have Ended's parody of that scene.
"My name is Khan..."
"Okay, I'm Captain Kirk, and this is Mister Spock."
"No, no, guys, listen to me. My name is Khan..."
"Why do you keep saying it like that?"