We only have one local rivalry. A local derby and a local rivalry are not the same thing.
Most dictionaries define a derby as "a football match between two teams from the same area".
Arsenal v QPR is a derby
Bristol Rovers v Yeovil is a derby
Barcelona v Espanyol is a derby
In each case they're not each others major local rivals, but that doesn't mean they're not derby matches.
FWIW - I actually disagree that Brighton is a derby match, but that's because I don't think 65 miles apart can be counted as particularly "local".
Saints v Bournemouth is a local derby.