BIN is a great product but it dries so quickly that you will find it (very) difficult to achieve a decent finish. It is also very hard so sanding it flat will take ages.
I would recommend Dulux Trade oil based undercoat (after priming). You may need 2 or 3 coats to obliterate the existing dark finish.
Mind-you, in spite of the above, if the doors are the originals you will probably find that the dark finish is masking years of scratches and dents. If that is the case then they will need a fair amount of filling with styrene resin filler (car filler, aka 2K filler, AKA 2 pack) and sanding, so BIN would be a viable option as the first base coat given that you will have a lot of sanding anyway.
If the existing finish is varnish the doors will need a decent sanding to provide a key prior to painting/filling, otherwise the paint will be prone to chipping. !80 grit silicone carbide (not aluminium oxide) will suffice.
The filler recommended is fairly hard to sand so don't go over board. Unlike the powered/ready mixed fillers, it can be sanded to virtually nothing (feathering). It is perfectly fine to reapply the filler over itself if the need arises. Depending on the temperature it can be ready to sand in 30 minutes, it has a working window of a few minutes so don't mix too much. DIY sheds sell it but it is much cheaper from car paint shops.
Apropos the finish coats, satinwood is slightly shinier than eggshell. I personally prefer eggshell, it also has the advantage of hiding imperfections slightly better than satinwood (which in turn is better than gloss).
Waterbased Vs oilbased- waterbased smells much less, dries much faster, won't yellow, easier to clean brushes but is more difficult to obtain a high quality finish, more difficult to sand flat between coats and is slightly less durable.
BTW if you do buy ZIN, also buy a bottle of household cleaning ammonia. You will need it for cleaning out the brushes. Just leave the brush soaking in it over night, it stinks but the ammonia evaporates after a day or two leaving water with white powder in the bottom. I cut up old milk bottles for pouring the ammonia in to.
Decent brushes will make life easier. The Purdy box sets are good value but you will need to find a proper decorators merchant. Avoid DIY sheds- they are often more expensive and frequently sell crap. Trade paints may cost more to buy but will often require fewer coats.
If you find it difficult to achieve a decent finish when painting then consider additives that will ****** the initial drying speed. Owatrol for oil based paints, Floetrol for waterbased and (possibly) a very small amount of denatured alcohol for BIN (I did once use vodka in a spray bottle, it helped a little but it has water in it).
Good luck
