The defining characteristic of fiat is zero (or effectively zero) cost of production. Hence, bitcoin is not fiat. "Commodity money" is a redundancy, as any tradeable good, including money, and yes proof of hash work, is a commodity. In particular, money is an exchange commodity.
I'm not arguing that bitcoin is not modeled after a commodity, but the "value" is virtual. If I hand you a paper wallet with an address and a private key on it, you can't use it for anything else other than scanning the key and spending it on something with a seller that will give you something for it in trade. With gold you can use it in building electronics that need non-corrosive conductors, oil can be used to run your car, rice can be eaten. Bitcoin's only value is that it can be traded for something of value which started when the first person ordered someone else a pizza for bitcoin.
You seem to be saying that there is no difference between a fiat currency and a commodity. That is one way to look at things, because you could argue that a post-Nixon US dollar has value as a commodity because the US government accepts it to pay your taxes and keep you out of jail.
You are right that both public coins and fiat currencies only have value as exchange commodities, nothing else. The difference is that the latter has zero cost of production while the former takes energy to produce. Fiat currencies can be issued "by fiat" in any quantity. Public coins are issued through an energy intensive and publcly monitored manner.
Personally I call everything a commodity that is traded for, but I'm not sure on the etymology of that word.