The name/title of Mandarin is never explicitly used in Iron Man 1, no, not that I recall. Someone can say otherwise if I'm mistaken.
At the close of Iron Man 3 Killian yells, paraphrase, "I've been in control the whole time, I AM THE MANDARIN"
What is "the whole time"? Does that encompass the 13yrs back to Bern that you were stiffed on the rooftop? Implying that indeed you did have a hand in anything 10 Rings related over that time?
Does it mean only since the Mandarin came on the scene? Which would be less than a year, cause he only emerges post-The Events in NY of The Avengers film as Tony makes mention of in the opening of the film when he 'turns on his tv'.
I'm not convinced they thought that out, the ramifications of what the 10 Rings/Mandarin change would mean to it in the big picture. Shane Black asked for a change and got it, Feige should've been the one looking at it more in context of big picture.
Could the 10 Rings group be pissed they've been co-opted, maybe? Could Killian be a ruse of a real Mandarin? Maybe? But to wheel out a Mandarin who actually
is a threat now, in this cycle of IM films, seems riskier cause most moviegoers have accepted the joke version.