I think of the government part as largely an accident; my definition of fiat money is “money that has value solely because the transacting parties say it has, and not intrinsic uses” - historically the way to get people to agree something intrinsically value-less has value was by government decree, but there are lots of exceptions. I think of the Yapannese rei stones as fiat money, for example (though there is some question about them having inherent value). Basically any money that isn’t commodity money (or directly redeemable for commodities) is a fiat money.