The definition of Fourth world
that I am familar with is not
"the worse half of the third world
The term Fourth world refers to nations that are not on the map: "internationally unrecognized nations". Peoples who still live, but their state has been submerged in another country. Or in many cases, the lines on the map divide an ethnic region into two or more "nationalities".
By that definition, areas such as Basque, Somalialand, Kurdistan are all fourth world. So are native American tribes. So are Wales and Scotland, though relatively developed and prosperous parts of first-world nations are not typical examples.
National boundaries are generally more recent and more arbitrary in the Third World.
The fourth world can be "the worse half of the third world" as ethnic minories are often neglected or opressed in the third world. Or in cases such as the Basques and Kurds, their desire for an ethnic homeland brings them into conflict with the existing states.