Global south and third world are different concepts, and cover a different set of countries. Brazil and Chile, for instance, were first world countries that are members of the global south. Here's a summary of differences between the terms:
Third World countries were countries not aligned with the first world (NATO) or second world (Warsaw Pact). The idea that third world nations were developing nations is a sort of neologism. (There were plenty of first and second world developing nations as well.)
Global South was a term used in the Vietnam war era largely academically, then introduced by the UN in the 2010s to avoid judgement around "developing" nations. (What is a developing nation? We sort of understand the term, but it's not self describing. Developing what, exactly?)
The global south is definitely a phenomena that shows in data, in general, the global south is much less wealthy, less politically stable, and there is less individual freedom. That much is demonstrably true, but the reasons for this are a subject of much debate. You might also debate the merit of using the term.
But if the set of countries exists, what do you name that set? Developing nations? That's probably not quite right, as several members of the set are developed. Third world? Also definitely wrong. Historically confusing to boot. I don't know what else to name that set, and I've settled on global south for no other reason than that's what others seem to use. It's an imperfect name.