There is an odd (to me, anyway) tendency in the US to use "village" to denote an area of a city. More recently, it's a marketing ploy to make somewhere sound more appealing, I would suggest. I live not far from a "village" which is nothing more than a new shopping development and devoid of homes. A few miles away is another "village" which is a housing development and devoid of shops.
In British-English, however, it denotes a small cluster of dwellings in the country/countryside which, although possibly self-sufficient (general store, post-office and eleven pubs, etc) is obviously not a town. I spent most of my life living in villages and never grasped the cut-off points where a hamlet becomes a village and a village becomes a town - or a when a town becomes a city, for that matter.
Over in the UK, if someone lives "in a village", they live in the country. Living in the country doesn't necessarily denote they also live in a village as they might live in an isolated home or farm.
I had never heard the Australian definition of "on country" before. Duly noted, but I suspect this might be unique.