Over-the Counter vinyl-free paint
Yes, it's called alkyd paint. Colloquially referred to as oil paint.  It is stinky, but good performing. It's becoming increasingly hard to find, as California and the EPA are pressuring paint companies to reduce the VOC content in architectural paints (i.e. the stuff sold at home improvement stores). Sherwin Williams continues to carry a good selection of alkyds. It's also readily available as marine paint. 

It's available in all the same "bases".  You know about the concept of "bases" of paint:


White base is heavily pre-loaded with white pigment. It's for off-whites. 
Medium base has less white pigment. It's for modest colors.
Deep base has no white pigment, and is for bold or dark colors. All the pigment must be added. 


My concern is that calcium carbonate is a very poor choice of pigment.  It is highly alkaline, like lye, and will surely damage the paint resin.  It's not a very strong pigment.  Titanium dioxide pigment is vastly superior, non-toxic, readily available, and will not degrade in sunlight (a rare trait today, as most color pigments are organic, and will degrade).   

So you should not be adding white pigment. You should select a white or medium base and taking advantage of the pigment already in the can.  

As far as slick, glossy surfaces, paint cannot stick to that.  I brush-paint some very aggressive epoxy stuff, and I basically ignore windows; I come back with a bottle of Windex and a razor blade and scrape it off the window effortlessly.  Paint does not stick to slick surfaces.   To make paint stick, you need to scuff-sand (remove gloss, not file material away).  My favorite scuff-sanding material is in your kitchen already; it's 3M ScotchBrite pads, green color.