The Chinese were producing spade coinage about the same time. Here is an example. These are cast instead of struck.

But yes, in the West, Lydia invented struck coinage. The American Numismatic Society has some good examples. The earliest finds come from excavations at the temple of Artemis in Ephesus, and look like they're coins minted by Croesus, so we can date that pretty well.

These are electrum (gold and silver alloy) and you'll notice the reverse doesn't have an image - just the whole left by the incuse die. It shows the art of striking coins in a very early state of development. As you know, later coins will be struck on both sides with images, including legends.

Putting your face on coins is a very late development. I'm pretty sure the first one to do it in the West was Alexander, and even then he was "disguised" as Heracles. The Diadochi transitioned into using their own images openly. In the Roman world it took even longer. Flamininus was the first Roman to appear on a coin, but it was a coin issued by a Greek state. Caesar was the first Roman to put his face on a Roman coin in his lifetime. Coinage is very conservative. It usually doesn't like changes.

That's not to say there weren't other forms of currency. Sumerians had clay tokens to indicate an amount of a particular commodity (maybe that's closer to a stock certificate rather than currency). The Chinese used shells before switching over to cast coins. Other civilizations used all kinds of things.

You don't really need coinage until you want a standard, regulated currency system, and that takes a certain amount of political development. If it's just you and I making a deal, we can measure out so many grains of gold in exchange for a cow or something. That's messy though. The point of striking coins - aside from propaganda - is to mark it the work of a trusted authority so you don't have to do all that measuring.

If you're interested in numismatics, ANS has all their students in their summer seminar read these before arriving.

C. J. Howgego, Ancient History from Coins (London and New York, 1995)

A. M. Burnett, Interpreting the Past: Coins (University of California, 1992)

They are very basic, easy texts that get into the basics of the history of coins (mostly in the West, unfortunately) and how to use them for history.

Lydia was the first to mint coins yes, but not to use a money. Almost every civilization had some good that was commonly exchanged, shells were very popular among all the coastal communities but there are many examples for many places.

/u/LegalAction covers it pretty well but there are a few key differences in Lydia coins than the earlier examples you mentioned. First of all, the Lydian electrum staters was stamped with the mark of a lion by the government making them more than just bean shaped piece of metal. The second was that the staters circulated among the common man. People used these coins to pay other people.

In one of my first AskHistorians post I actually went over the evolution of faces on coins.