You're right it is weird, and its also a bit complicated.
Its on multiple levels but its first and foremost to distinguish between Nottingham Forest and Notts County. Locals refer to Notts County as just Notts so Forest didn't want that word association of their nearest rivals. It would be somewhat like having the word Liverpool in the Everton name somewhere. The same for County as their fans dont want to be known as Nottingham County although very few ever calls them that.
Geographically it doesn't help matters that Forest now play in Notts at the City Ground, and County play in Nottingham at Meadow Lane, so their names are actually the wrong way round.
Note:
Nottingham Forest did originally play within the City of Nottingham when they were first formed
Further Note:
Derby are sheep shaggers though .. that is indisputable!