This is just a BBC/media thing, based on nowt except some sort of elitist "world brand" agenda (except Brighton and Palace, but they're London-ish so got to stick them in, AND they're in the PL).
A proper derby should be local (ideally in one town or city, but close-ish if not).
It should be so passionate and stressful that the only pre-match sustenance you can keep down is beer. This also acts as an anaesthetic.
On the day, there has to be an irrational hatred of your rival so strong it hurts (and will be unbearable if you lose).
It should be played in a totally febrile and manic atmosphere created by people who are utterly taken over by the whole experience and who spend 2 hours on the verge of a complete breakdown.
It doesn't matter what division it's in, these days ideally NOT the PL.
People who chose their team by sticking a pin in a PL league table need not apply.
By these measures, there aren't actually many real derbies, and some only happen rarely.
I think these are "real" derbies:
Us and Them.
Blues and Villa.
Wolves and West Brom.
Stoke and Port Vale.
Bristol City and Rovers.
Portsmouth and Southampton.
Forest and Derby.
Blackburn and Burnley.
Clearly there are others (Newcastle and Sunderland, Norwich and Ipswich get honourable mentions), but I've only tried to include ones where the feeling is raw and visceral, I've no time for PL niceties, or that Liverpool/Everton thing where they allegedly sit together. That's not a derby!!!