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I can usually tell whether paint is oil or latex by the feel of it... oil paints tend to be very smooth to the touch while latex paints have a more rubbery feel. The difference is more distinct with gloss paints, less so with flat paints.
Of course, this method may work for me but is not very helpful if you don't have a tactile "frame of reference"... e.g. experience. Since your objective is to paint over the old paint, the easiest way to tell is to rub an area of the paint with either denatured alcohol or a paint deglosser such as Wilbond. If the paint is latex, a small amount of paint may be removed and/or the paint surface will become slightly tacky. If the paint is oil, neither of these things will happen.
The tackiness will normally disappear within an hour. If the paint has a gloss or semi-gloss finish, the gloss will be diminished in the treated area. Paint deglossers are also known as "liquid sandpaper". By removing the gloss (and also cleaning the surface), they allow the next coat of paint to adhere more firmly with no-or-minimal sanding. Oil paints, especially glossy ones, should always be lightly sanded unless they are primed with a tenacious primer such as Bin (shellac), Kilz (oil-based), or Zinssner 1-2-3 (water-based).
In summary, there are three rules when painting over an unknown surface... sand, clean and prime. Follow these and you will probably never suffer the misery of paint failure again!
Painting cement board is not really a standard practice, but it can be done. Seal the seams with standard wallboard tape and compound. Since the product does not have the tapered edges of wallboard, you must feather the compound out at least a foot on either side of the joints to give the illusion of a flat surface. Use a water-based primer/sealer first, followed by a coat or two of a mildew resistant bathroom paint. (A washable eggshell or semi-gloss latex paint with a mildewcide added would be my distant second paint choice.)
If you find the finish on the cement board to not be to your liking, you can always use a texture paint to mask any defects. Then apply two coats of mildew resistant paint to seal and protect it.
For future reference, water-resistant "green" wallboard would have been a better choice for a painted surface. It is designed to be finished like regular wallboard, but withstands much higher moisture levels without damage. There has been some controversy about this product, and most tile contractors no longer use it within shower and tub enclosures as a backer, preferring the various cement-based tile backerboard products like you used.
Being a curious-cat, I decided to do a simple product test of water-resistant wallboard a few years ago. I took a 6" x 6" piece, placed it into a large pan of boiling water and let it boil for an hour. The result… there was no deterioration in the paper whatsoever (though it did absorb water), and the solid core showed not the slightest bit of softening. So the question that arose in my mind was… why is this product frowned upon?. Though not a complete answer, I realized the sad truth was that contractor error or irresponsibility could be part of the problem, not the wallboard. I have done at least 50 tile repairs in fairly new homes… less than 12 years old… caused by water leaking behind tiles in enclosures. What I have found consistently is that the contractors used regular wallboard, not water-resistant wallboard, in these failed jobs! I have personally never seen water-resistant wallboard properly installed on walls or ceilings fail due to dampness in residential conditions.
Yes, there is. Exterior paint contains certain fungicides and UV protective additives that are not approved for interior use. Because of the extreme conditions exterior paints are designed to endure, the strength of the preservative chemicals used may be unhealthy in closed inside areas.
This begs the question, "Why do you want to use exterior paint inside your home?" If you are looking to get some of the qualities of exterior paint, such as mildew or water resistance, there are interior paints and additives that can do the job just fine.
For example, there are special bathroom and kitchen paints that have very durable, eggshell finishes plus tremendous mildew resistance. There are also additives, known as mildewcides, that are designed for interior use. They can be added to any paint, even flat ceiling paints and texture paints, to make them more "bathroom friendly" and mildew resistant.
The most durable paints are the most glossy, as a rule. Gloss paints tend to be harder and less porous, meaning they resist moisture better. Usually, a glossier surface is preferred when you anticipate the surface will be cleaned often or have regular contact with hands or other objects.
For these reasons, moldings, trim, bookshelves, doors and windows are usually painted with semi-gloss or gloss paints. However, there are also paints that are called "satin" or "eggshell" which are somewhere between semi-gloss and flat in appearance... just a touch of sheen. They are often used for entire homes... trim and siding... with success. They are not quite as tough as a semi-gloss or gloss, but have the advantage of being more washable than a flat finish while not showing up every surface imperfection as the glossier finishes will.
Then there is the eternal question… latex or oil paint? My choice for garage doors is oil-based paint for reasons related to long term maintenance. Oil paints can be sanded more smoothly than latex paints… an important quality for any surface that may be repainted often.