The terms are ambiguous, and their general definition shifted considerably after the cold war. Originally the three worlds referred to the cold war alliances, with the first world being the US and its allies, the second world being the USSR and its allies and the third world being the neutral countries. In the following map, third world countries by the cold war definition are shown in green:

Switzerland is a typical third world country, isn't it? ;)
The cold war definitions became irrelevant after the collapse of the Soviet Union, and even more so as traditionally communist countries shifted to a more Western model. After the cold war the definitions shifted to reflecting the "worlds" broad economic status, and by that definition many of the cold war adversaries, including both the US and Russia, would be placed in the first world.
The change in definition, from political ideology to economy, contributed to the gradual disuse of "second world" that by the later definition would mean countries that are somewhere in between poverty and prosperity. Without as clear a line between first and second world countries, the term became more confusing than accurate.
Developed and developing country and emerging economy are also vague and broad terms, but their definitions are constant and don't carry the no longer relevant cold war baggage.