As an Australian, I do find it a bit perjorative for countries north of us (many of them in the northern hemisphere!) to be deemed the "global south", while we are excluded despite actually being the only inhabited continent entirely in the south. It just reminds one that nobody cares about the southern hemisphere, and that northern hemisphere types think anything south of the mediterranean is "south".

North/South doesn't have anything to do with it, anyway, as you alluded to. What people actually want to talk about is whether a country is a former colonial master, a former settler colony or a former extractive colony (or possibly multiple of these, as with e.g. the US).