So what exactly is a "THIRD WORLD COUNTRY"?

You seem to have the right idea. I think that there is something a bit elitist about using numbers to refer to the development of a set of countries. We have to be careful about who we could be offending with these words, and unless you are someone who grew up at the time of the World Wars, and the definitions still make sense to you, it feels more polite to use "developing countries" rather than numerical orders as a description, as you say.