I was told today that here in Australia I should write a dollar sign ($) with only one line through it and that a dollar sign with two lines is American. Is there any truth to this?

That is an urban legend.

As an American, I always write my dollar signs with just one slash. However, I am familiar with the two slash dollar sign and found out this mistaken theory about it.

That $ is a monogram of U. S., used on money bags issued by the United States Mint. The letters U and S superimposed resemble the historical double stroke "$" sign: the bottom of the 'U' disappears into the bottom curve of the 'S', leaving two vertical lines. This theory, popularized by novelist Ayn Rand in Atlas Shrugged, does not consider the fact that the symbol was already in use before the formation of the United States.

https://en.wikipedia.org/wiki/Dollar_sign#From_.27US.27

Save energy and use one line. The standard keyboard on computers used here in Australia has a single stroke. (reference - I'm looking at one right now.)

Both the single stroke and the double stroke are acceptable as symbols for dollars. There is a similar symbol called the cifrano (S with two vertical strokes, running from the top of the S and extending below it, not centred vertically) which is used to represent some South American currencies, such as the Brazilian real.

https://en.wikipedia.org/wiki/Dollar/Peso_sign

https://en.wikipedia.org/wiki/Cifr%C3%A3o