Since Switzerland, Sweden, and Ireland were all considered third world countries by the original definition (not allied with either NATO or the Warsaw Pact), it's safe to say that first and third world country are not very rigidly defined. Even at the time, the original definition was pretty creaky since alliances were rarely explicit and the polar nature of Soviet vs US interests made cases like China after their break with the USSR hard to place.
Labeling countries as third world as shorthand for impoverished, or undeveloped stems from the US and Soviet tendency to use developing non-allied nations as battlefields in their proxy war (ie Vietnam, Afghanistan, Egypt) in order to avoid directly challenging each others' spheres of influence. Since the competition to expand influence in the "third world" ranged from bribery through foreign aid to overthrow by force, the contested countries naturally began to look largely the same to the general public- poor, war torn, and unstable.
So-- obsolete term with pejorative overtones that just needs to be retired altogether.