Exactly this. Notts is short for Nottinghamshire i.e. the name of the county, hence Notts County.
Nottingham Forest are named after Nottingham the city rather than the county.
(should have called themselves Nottingham City to prevent confusion)
Indeed.
I mean, a forest is to be found in a county (or shire). It is never in a city.
One of my best mates is a Forest fan from Nottingham and he cracks up when people refer to “Notts” Forest