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I have garnered a reputation for recommending Zinsser & Toupret! :) I'm looking for UK based alternatives but I let the rest of the trade do most of the job and then try the ones that are popular with a no nonsense decorators merchant in the local area. Plus a lot of discussion with other tradesmen over the years to find out what they like to use.
I've used pretty much everything that Zinsser make and they are a decent boot and braces option for most people but I do also try out more exotic paints for different jobs/reasons.
You're right cover stain is oil based and it's available in a spray version for touching up stained poly tiles in ceilings or just stained ceilings if you don't mind covering everything in the room. It's also necessary in some cases for blocking the stain from exterior pine if it's been completely stripped of paint, the other primers tend to let the stain bleed through.
BIN is the only one I don't use much these days because it needs meths clean up and I try not to keep white spirit and meths on site.
If your finish is as good as every coat you put on :) It's why I'm always banging the drum for Gardz, it was one area where everyone has their methods and I've tried loads but when you have to fill a crack running right through a white ceiling then re paint the ceiling it's the only product I've used where you will not see the repair after 2 coats of white paint. Of course getting a decent finish on ceilings is an art anyway.
I think Cover Stain is alkyd/oil based.
BIN is shellac based - it’s the only Zinsser product I’m really familiar with, I always have a tin.
Either would be fine, I think oil or shellac based primers are still the best thing to use, I often use aluminium oil based primer on hardwood.
Your finish is only as good as your first coat, something something.