if you put them on a line of increasing carbon content, they actually overlap. you'd have elemental iron on the left, then wrought iron, and then partway up wrought iron you'd also have steel. and then just steel for a while, and then overlaps with cast iron.

wrought iron is distinct from low carbon steel due to the inclusion of slag in wrought iron. wrought iron is a very crude alloy. that gives it some properties different from low carbon steel, but its real virtue is how comparatively cheap it is to make. not having to worry as much about the impurities means less processing.

the difference between very high carbon steels and cast iron is a bit harder to explain, but has to do with how the carbon is tied up with the iron. cast iron has free carbon in it where steel doesn't.

as for "why is the naming convention so screwed up?" we discovered them out of order. cast iron is just barely iron in a meaningful sense, wrought iron is early man's best attempt at purifying iron completely. landing on the sweet spot in between is really hard, in no small part because while you're trying to make the steel behave, it's trying to rust and also set everything on fire.