The etymology section of your link suggests different:

The demographer, anthropologist, and historian Alfred Sauvy, in an article published in the French magazine L'Observateur, August 14, 1952, coined the term third world (tiers monde), referring to countries that were playing a small role in international trade and business. His usage was a reference to the Third Estate, the commoners of France who, before and during the French Revolution, opposed the clergy and nobles, who composed the First Estate and Second Estate, respectively (hence the use of the older form tiers rather than the modern troisième for "third"). Sauvy wrote, "This third world ignored, exploited, despised like the third estate also wants to be something."

But you're right in that the term began to be used far more widely during the Cold War for political alignment.

I thought it was a classist term