Historically no one seems to know, how it got its name or who gave it. It didn't have a name in his manafactory seems to be clear as his correspondance with his partner confirms, always using the term 'engine', never Rocket. Then it seems after it was dismantled and shipped down and rebuilt for the trials did the name Rocket appear.
The top picture I think is the Mk2 version of Rocket as the cylinders are set at 8 degrees instead of the 35 in the second image, this gave a smoother drive to the wheels.
Unless someone knows differently, the naming of it is lost in history with little or no fanfare........Stephenson didn't like blowing his own trumpet....!