I've heard that tall tale too, but point in fact it was Honda that turned him in. He and Coughlin went to Honda for a job interview with a CD that contained all kinds of data. Stepney had said that it was his body of work along with Coughlin's. Honda was very suspicious about the whole affair and blew the whistle.
It of course turned out to be data from not only Ferrari but McLaren as well. Although McLaren was the only team punished for it when only Coughlin knew anything about it.
He didn't photocopy a damn thing. He just downloaded it all to a CD and walked out the door with it.