Any country not aligned with NATO, Capitalism, or Soviet Communism. The term was misused and misunderstood, and due to perceptions on the living standards of other countries and its frequent context as a prejorative many simply assumed that not being aligned to one of those powers meant poverty. So here we are post Cold War where the term is largely a historical artifact or specialized use case, but has lived on due to misuse.
With a per capita GDP of around $30,000 USD annually than the US Bermuda could be considered a "rich" third world country. Of course it's a bit of a trick question as "rich" is ill defined. For instance by the end of the Cold War China was a third world country and is very wealthy, but the distribution of wealth is not equitable amongst its population. Similarly there are third world countries with a low GDP or per capita compared to some first world countries but that still boast either an overall high standard of living and satisfaction or an elite wealthy class. It's all sort of relative based on how you define "rich" and debatable as to what you would call a third world country since many countries changed sides and leadership over the decades, have undergone economic changes, and few countries weren't in some way at least by proxy helping or taking aid from one "side" or both. The UAE for instance could be classified as "third world" since its an absolute monarchy.
It's a political term.. not Financial !!