In terms of sub-variants of lead white, the pigment used for modern lead white is not made in the same way as the historical pigment for lead white. One of the differences is the particle size and shape of the pigment itself. The historical Dutch Process, sometimes called Stack Lead White (not shown here) is harder to find than the regular lead white and is favored for this different handling. Stack Lead white has different properties in its oil paint behavior (the five-dollar word for this is thixotropy).
As with any other white paint which will be mixed into the other colors on the palette, binding oils make a difference. It's common to find Lead White in linseed oil or walnut oil. There are many names for lead white, which unfortunately have been used rather freely by paintmakers over the years, so it is often hard to determine exactly which name refers to which pigment variety or blend. Sometimes within a brand they will have a Flemish White, a Lead White, and a Cremnitz White, and they will use these historical names to differentiate between lead whites made with different oils, or different pigment blends.
Lead White is also known as Flake White, Cremnitz White, Cremser White, Dutch Process White, Ceruse, and Silver White, to list just a few of the most common names.
Sometimes these names referred to blends of lead white with other pigments, such as zinc or barium sulphate. For example Rublev/Natural Pigments offers a handful of lead whites which have different combinations of pigments or binding oils, as does RGH.