That's the looser definition, the narrower one (from which the name comes) is that that it is currency not backed by collateral but backed by a government's designation of it (by fiat) as legal tender.

Given the absence of significant non-representative non-government-issued currencies prior to modern cryptocurrencies, the looser definition was essentially equivalent to the narrower and naming definition, so there was really no real reason to concern oneself with the difference. Now, though, especially in the context of discussions of cryptocurrencies, its probably better to stick with the narrow definition of fiat to avoid confusion and adopt a different term when something like the broader definition is necessary.