"Steel Alchemist" is in fact the literal translation of Edward Elric's title. The term "Fullmetal" is most likely a reference to Stanley Kubrick classic war-is-hell film Full Metal Jacket.
The "jacket" of a full metal jacketed bullet is usually made of steel. Sometimes you hear these bullets called steel jacketed in English, and in many other languages, a translation of this second name is much more common. In many countries outside the US, FMJ was retitled to reflect the most common local name for the bullets, in order to preserve that reference. I don't know if Japan did this: IMDB doesn't list the Japanese title for some reason.
But if Japan did retitle the movie, then the title "Steel Alchemist" may have been intended all along to reflect the title of the "Steel Jacketed" movie. Many translations of FMA actually do use a title that reflects the way FMJ was titled in the translation's own language, and so it's possible that "Fullmetal Alchemist" is just another application of that practice.