I can believe that there are variations in terminology between the states (as one of the sources suggests). Maybe this is what I'm misunderstanding: does the Line run along state boundaries? (the article suggests not) The sources suggest that the variation is defined by what state a person is in, not on which side of the line they find themselves.   

No, not purely on state boundaries. It's remarkable that the straight line on the map does the job so well. Of course, that's an approximation too. The exception to state boundaries is in southern NSW, mostly the Riverina, which is Aussie Rules country. One can see it by looking at the shapes of footy grounds and goalposts when travelling through that area. (Even on Google Maps, if you want to do your own research.) This is probably due to the fact that it is actually closer to Melbourne than Sydney, and the early rail lines connected to Melbourne rather than Sydney back in the nineteenth century. Wagga and Narrandera are boundary towns. Broken Hill is an Aussie Rules town, actually being closer to Adelaide. (Did you know that Broken Hill runs on South Australian time?) Of course, journalists aren't going to go into that sort of detail every time they write about it. The line is wrong for the Northern Territory. All of the NT is Aussie Rules territory.