The term 'derby' is used to refer to the big event, originating from the Epsom Derby. Typically, this would be a local derby as Everton's big game would be against Liverpool and Man City's would be against Manchester United. However, the game against the main rival would be the better definition.
Please, put this on a post-it note on your computer screen, particularly if you are from Hull or Lobbygobbler.
I'm not too sure about the wikipedia entry. It was my impression that the first Derby match in any sport was between Wigan and St Helens as the Earl of Derby's estate covered both towns. So both the horse race and the local match are independently named from the same source.