First, I don't accept the premise that "steel" exists as distinct from "iron" at this time (see link below). But that's a diff discussion, so I'll just link to that and address this.
(Here's an older thread on "steel" that some may find enlightening re the question "What is steel in the middle ages?" - here ya go:
As with ghosts, spirits, elementals, undead, demons and so many other magical "types" of creatures, imo it's better to shy away from blanket rules for all of them, or at least not when it's a "problem solver" for the PC's. If cold iron is death to some delicate fae but merely enrages some of the more surly types, your players will find more challenge and the story will be better for it in the long run.
I've always found the image of a stroke (the brain hemorrhage type) being caused by an invisible faerie dart to be a strong one. Piss off the wrong fae, and that's what happens.
The problem is that cast iron simply didn't exist (in the west) until the 14th-15th century, but the phrase "cold iron" certainly existed, and from all appearances not just in a poetic form. (That is, it was seen as distinct from the default form of iron, and not just called "cold iron" because all iron is cold to the touch.)
So clearly, "cold iron" and "typically-produced iron" are seen as different things in the medieval mind (and casting is not the distinction).
(Yes, magic could easily melt and cast iron - but the distinction of the term "cold iron" does not take that possibility into account. At least, not in RL - as always, ysmv.)
So "cold iron" is called that because it's distinct from other types of "iron". I always had the distinct impression that "cold iron" was (mostly) cold forged, hammered out with very little heating, an archaic and very difficult process. (I'm not sure how else it would be distinct.)
Steel is certainly mostly iron in a chemical sense, comparing modern elements, but the middle ages did not deal in "chemicals" or "elements" of that variety. They dealt with things as they were produced, as they appeared, as they were perceived. And "iron" and "fire" were the two central elements (for lack of a better term) that were involved in making steel.
(Either way, steel, by necessity of its production, is iron that then has gone through repeated heating processes, and so, while still "iron" in content, is not "cold iron" by any definition.)