Why we are using organic zinc rich paint for the priming of stainless less steel, galvanized surface etc. and inorganic zinc rich paint for carbon steel metals ? What is the chemistry behind it ?

Carbon steel corrodes very easily. Zinc (metallic zinc, not the ion) is an active metal. In the electrochemical series, it is more active than iron. If something attacks the underlying metal, electrons flow to the zinc and, because of its reactivity, the zinc reacts (converted to Zinc +2) instead of the underlying metal.

Galvanized surfaces are coated with zinc. They have the same properties.

If the "organic zinc rich" paint truly has "organic" zinc in it (more properly organometallic), it does the same thing, but I tend to doubt that it does. Organometallic zinc is generally too reactive to exist in air very long.

I have heard that there are some companies (not very reputable ones, IMHO) that will simply put zinc salts in paint and call them zinc rich. I haven't verified this, but if it's true, that would be a pretty poor excuse for a "zinc-rich paint."