The two widely available pigment colorants for oils and varnishes are oil colors and Japan colors. The difference is that oil colors are pigment ground in linseed oil while Japan colors are pigment ground in varnish.
So the difference in practice is that Japan colors dry faster and harder than oil colors, though if you were to mix an oil color with varnish, it should dry well.
The name “Japan” comes from the attempt in the West to imitate Japanned furniture (also called Japanese or Oriental Lacquer) that was imported in the 17th and 18th century. So the harder drying and glossier varnish base worked better than an oil base.
In my experience you can also use widely available universal colorants (normally meant for tinting latex paint) in oils and varnishes successfully if you let these colorants sit in the binder overnight while stirring occasionally.