B-I-N and Cover Stain are slightly different products. They are both oil based so you need thinners to clean the brush. B-I-N is more often used for sealing wood, I don't use it much but it's a lot thinner than Cover Stain and needs cleaning with meths if my memory serves.
Cover Stain appeared after 123 and B-I-N had been around for a while. I had a lot of success spraying Cover Stain and matching existing ceilings, they make a spray can that sprays vertically but the overspray is very fine so anything in the room needs covering to protect it. That's why I'm reluctant to recommend it. Brushing 2 coats on the stain and feathering at the edges or rolling and feathering with the roller if you can manage often works quite well.
I would guess Cover Stain is around British standard white, rather than British standard brilliant white.