"You said you wanted the Mandarin,
your looking right at him. It was
always me Tony, right from the start!"
In Killian's mind:
The Ten Rings is real
The Mandarin is their leader
The Mandarin is just a myth
The Ten Rings was extinguished after the events of Iron Man (Stark and Stane seperately playing their parts in their "demise")
He decides to use this "fallen" terrorist group to control the War on Terror
He hires Trevor Slattery to act as The Mandarin, thus "creating" The Mandarin
Because he "created" The Mandarin (as is his belief), he in fact IS The Mandarin.
Much like Bill Gates created Microsoft, there for IS Microsoft. He "created" him, so he has the rights to call himself by that title.
In Reality:
The Ten Rings is real
The Mandarin is their leader
The Mandarin has led The Ten Rings for an unknown length of time (years, decades, centuries)
After the events of Iron Man, The Ten Rings lied dormant (as other posters have said), recovering from the hit they took
Killian hired Trevor Slattery to portray The Mandarin
The Mandarin saw Trevor's "lessons" to the US and took offense to their disrespect and impersonation of him and their organization
After the events of Iron Man 3, The Mandarin and The Ten Rings have finally learned that it was Killian who spearheaded the facade
With Killian dead, Trevor was the only person involved left to punish
Cue the events of "All Hail the King"