You're using an old defintion that is very rarely used today. It's now considered archaic, relegated to the dustbin of history. In fact, you're attempting to educate the parent precisely because of how old and out of use that defintion is. Further, it's so out of use, the most common dictionary references to eg first world, are always about level of development / industrialization and not cold war political alignments.

First, second and third world is now drastically more commonly used as a reference to the level of development.