That depends on how you define fiat money. Fiat coins are as old as the Chinese empire. Even in Europe, many coins were significantly underweight. However, the theory was always that the coins contained their worth in metal. In practice, that was not often the case, though.
The gold standard described above was itself derived from the silver standard and the gold/silver standard. Its lower denominations were fiat coins, but in theory, you could always exchange them for a full value coin. Therefore, the value of those coins lay not in the trust that even though lightweight, each coin contained enough metal to give some security but in the trust that the government/central bank had enough metal to redeem all money into metal.
If you grasp that evolution, you will see that Bretton Woods is but a small step further in that logic. After the second world war, the US was a dominant economic power, dwarfing all others. It made sense, then, to replace the trust in your own government by the trust that your government had enough credit with US government, which in turn had enough metal to cover their debt. The US was the holder of metal reserves of last resort for other countries. If you wanted your cash redeemed in metal, you would in theory go to your local central bank or government office, who'd pay you in dollar notes, which you could offer for payment in the US where you'd get your metal.
As the gap between the US and other countries got smaller and the Vietnam war made the amount of USD outstanding in other countries larger, the US's position became more and more difficult. It became untenable when oil prices shot through the roof, so the US cut the theoretical link between dollar notes and metal. Nothing happened.
At what point do you see a change to fiat money? Around 1700, when silver hunger led to the first experiments with paper money? Around 1816, when lower denominations had to be exchanged for full value coins? After 1944, when the ultimate responsibility for redeeming cash in metal came to lie with the US? In the seventies, when the last silver coins were demonetised? Or did we always have fiat money, but we chose not to think of it as such?
Peter
