I make a lot of wooden games for the kids in our street to play with. My neighbour, who is a painter, paints them for me. He uses ordinary white paint for the first coat, then adds the top coats of varying colours. The depth in the colour is a lot better using the first white coat.
As far as child safe paint, is there such a beast. I don't think so because most of the pigments come from the earth as well as man made products, and unless you know what they consist of, then consider them to be a possible hazard. Take the humble Talcum Powder (used as a pigment extender) This has just been tagged as unsafe to use on babies, Johnsons are now in the spotlight over this issue.
The worst component used in paint many years ago was lead, it was used in a liquid form as a dryer for the paint. This is no longer used in the western world and has made paint more safer for all people concerned.
As for paints made in China, right now, I would not trust them at all. The strict guidelines set for the western countries do not seem to be in place for a large range of products coming from China. I would recommend that you buy locally made paint and steer clear of Chinese made products.
After working for a major paint company here in Australia for 22 years, I made sure I read all of the MSDS sheets before handling any of the products. That includes the humble Talcum powder.
There is plenty of information on the internet and after some searching, you will find the manufacturers MSDS (Material Safety Data Sheets) for the products they use.
Dave.