Just to be clear, a rivalry is still known as a rivalry in England. A derby is a game between two rivals, such as the Merseyside Derby between Everton and Liverpool. And 'dar-bee' is just a pronunciation of the word derby used in some parts of England.
That said I'm very much looking forward to the Hudson Derby (pronounced der-bee, because that's how I say the word usually).
Just to be clearer, a derby in the UK is a match between two geographical rivals, either in the same town or city as each other or very close by. No-one here, for example, would refer to a game between Man Utd and Liverpool as a derby, even though they are only 30 miles apart, because the rivalry was formed through competition, not because they are near each other. Exceptions are made for clubs without a nearby rival at the same level, hence why Newcastle, Sunderland and Middlesbrough's are derby rivals despite each being the only professional club in their city. The three cities are nearby to each other, so they are automatic geographical rivals.
The caveat to this is that if you are based nearby another club you are deemed rivals regardless of past history or dispute between clubs. If, by some small chance, a non-league London team in the 10th tier of the league pyramid were to get far enough in the FA Cup to play Chelsea then it would be yrunpted in the media as a derby.
This, for the record, is why with no prior justification I have always been quick to throw insults towards NYRB and the Cosmos (especially back in the earlier days of the forum, for those of you who were around to remember that) and yet have never once talked about the Revs or the Union as potential rivals. In the English system, they are simply not close enough to be considered rivals.
