Yes, you've almost got it!
[Actually, Barnsley's stations were operated by the L&Y [Exchange] and GC [Court House].]
All four had competing stations [now closed], belonging to other railways.
What were Barnsley Exchange, Rugby Midland & Nottingham Midland had competition from the Great Central/LNER - Barnsley Court House, Rugby Central & Nottingham Victoria.
Wolverhampton High Level was the odd one out as it had competition from the Great Western at Wolverhampton Low Level.
But I think you're near enough.