I'm sure after the process of being booked, Rush was able to leave on his own recognizes. This was a deal.
Defendants are often let go by the police on their own recognizance after arrests, voluntarily submitting themselves after a warrant is issued or not. It's still an arrest.
For an arrest to occur, it does not have to entail the cops coming after you, putting handcuffs on you, and incarcerating you.