I wouldn't experience the arcade version until over two decades later, so nostalgia has me side with the NES version (didn't know any better back then ).
As for the Radical link, I've read "skate-or-die" wasn't a term used in skateboarding until it first showed up in 720* (EDIT: not true), which is the inspiration for the Skate or Die series (true), and I presume the use of it in Radical (probably true ).
The Skate or Die cover art/logo is also mimicked on the side of Guns 'n' Roses LE, and the Radical logo seems to take cues from it.