I think that the term "wrought" refers to making the iron by mechanical means, i.e. hammering from a bloom. So, "wrought iron" would be foremost pure or nearly pure iron, and secondly "wrought" or produced by hammering. A given puddled iron could be as pure as another wrought iron but have been produced by another method, thus not truly being "wrought."
It is true that in the nomenclature of common ferrous alloys, the metallurgy, manufacture of, and common name of an alloy may dovetail well, but it's not necessarily ironclad.
Today though, it does seem as though the term "wrought" encompasses nearly all iron alloys with less carbon than mild steel, "commercially pure" being an exception.
The history of iron is fascinating- it used to be a much more dynamic field, full of mystery and regional peculiarities. I always recall Vannoccio Biringuccio referring to a certain iron ore as being "wild and sour." Intriguing to think of a time when vocabulary such as that was used to describe iron! A thing so commonplace as to be beneath the attention of most, these days.