There are actually three "Three Worlds" models in history that you can talk about here.
Karl Popper's 1978 model is a lense on reality rather than a political concept, dividing the world into physicality, mentality and knowledge. It has little to do with the geopolitical frame you are referring to, but Popper referenced it in his work, which makes this an interesting footnote.
Mao Zedong developed a "Three Worlds Theory" (三个世界的理论), first introduced to the world stage on 12 April 1974, two years before Zedong's death, by Deng Xiaoping in a speech to the United Nations. This in fact is a geopolitical concept dividing countries into three worlds, mainly based on wealth. The US and the USSR were the First World, their allies were the Second World, and the poor countries in Africa, Latin America and Asia (which was the entirety of Asia except for Japan) were the Third World. This model provides a fascinating reflection on the Chinese self image at the time, as it was a bitterly poor country at that point and already hostile to the USSR after the 1960s and the Sino-Soviet split. China at that point already had possessed nuclear weapons for a decade (1964/1967 being the landmark nuclear firsts for the PRC), and it hardly was a minor footnote due to its nuclear potential - but it needed to appear that way to put itself apart from the Soviet Union and the United States.
Finally, the western Three World Model. French demographer Alfred Sauvy initially drew a comparison between the two power blocs of the Cold War, US and USSR, to the two ruling estates in pre-revolutionary France, the nobility and the clergy. Meanwhile, he likened the unaligned states - most of Africa, South America, Asia - to the common people, the third estate of said pre-revolutionary setting. It is of course a bit of a long shot, but it serves as a decent enough transition into the pure geopolitical meaning of the term that you are most likely thinking of. You see, whereas the Maoist model was very typically communist in its focus on wealth as a dividing factor between the nations, the western model was much more diplomatic in nature - the world you were in was determined by the diplomatic alignment of your government, not even so much the government's guiding principles and values. Apartheid South Africa for example was a First World Country, whereas Austria and Switzerland were part of the Third World due to their non-alignment. Now, if we are to ascribe values of freedom, democracy and somewhat libertarian non-autocratic government to the western "First World", then surely Switzerland and Austria would have made a better fit than South Africa. But that is not what this western Cold War model was about, it was purely focussed on Realpolitik and on diplomacy.
Now, you are probably thinking of the western model described just above. However, the "worlds" as terms have been bastardized since the end of the Cold War. Few people still use it as a diplomatic alignment chart between NATO and its geopolitical adversaries in Moscow and Beijing.
The term "Third World" nowadays is most often used synomously with "Developing Countries", referring to poor and downtrodden places like many countries in Africa as well as places like Haiti and Afghanistan. Now, most of these countries would indeed have qualified for the old diplomatic meaning of "Third World" as well, but nowadays, the division into worlds is much more economic in nature, although I would not go so far as to conclude that journalists and authors have all suddenly become enamoured with Maoist literature and started using Zedong's political models. Whereas "Third World" often just generally means 'poor' and "First World" can be used as 'rich', the "Second World" as an in between ground has been lost in this modernized changed meaning of the terms. This already hints at the fact that we are no longer using the terms like Zedong or Sauvy originally intended them to be. Djibouti for example would be a "Third World" country economically, but due to its close military relationship with France could argued to be "First World" in Sauvy's non economic model.
I would personally conclude that it would be best to resign the Three Worlds Models to the dustbin of history and to properly refer to poor countries as LDCs or similarly appropriate terms.