Aussies - whats the origin of "beanie"

Factual Questions
1

I think beanie is a unique Australian name (for a woolly hat)? Anybody know the origins behind this? Is beanie an abbreviation for something?

2

According to the Macquarie Dictionary, which is the definitive work on Australian english, it comes from the colloquialism of “bean” meaning the human head.

3

I associate beanie with the US - specifically with kid’s propeller ones and students. This document on skullcaps, that I’ve only skip read I must admit , supports that.

Doesn’t mean that the word originated there, but the skullcap guy lists it as US headwear.

4

Yes, the Macquarie is a complete dictionary of Australian English, and not a glossary of slang. It may indeed be an American word originally.