The Lancashire & Yorkshire Railway historically operated only one Club Carriage. Carriage No 47 was a bogie vestibule 1st class vehicle for the Blackpool to Manchester service. It was built in 1912 to replace two earlier arc roofed coaches. It was 58 feet long and 9 feet wide, and was fitted with ten foot wheelbase wide bearing bogies.
Wide picture windows were installed, developed from the L&Y dining cars and railmotors. The coach had only one central door per side. In around 1917 it received some minor alterations at Newton Heath Works, including the fitting of electric lighting, possibly to provide better conditions for playing cards.
It seated only 40 passengers; the 40 members of the Club, an organization of Manchester businessmen who lived in Blackpool and travelled to work on the same train each day. By agreement between the Club and the L&Y, the carriage was always marshalled as the second carriage in at the Manchester end, on the 8.10am from Blackpool and returned on the 5.10pm from Manchester Victoria.
It was, in every sense, a Gentlemen's Club. Membership was by elections, was restricted to a maximum of forty, and had its own rules. The carriage was fitted with armchairs, 30 of which were for smokers. It also had occasional tables and an attendant to serve tea to the members.
The so-called "club trains" referred to in the OP were formed of a motley collection of mostly Mk I and MkII coaches in every livery imaginable hauled by Class 31 or 37 on a small number of Regional Railways services for a short time in the early 1990s. They were pressed into service because of a temporary shortage of 2nd generation DMUs.
I don't know who started calling them Club Trains, or when. I don't recall hearing it being used at the time. The epithet was either applied by someone with no knowledge of what the Club Carriage actually was, or by someone who was fully aware but had a keen sense of irony.