It’s more the other way around: Iron with more than ~2.1% carbon is brittle, and therefore it cannot be worked with tools; it can only be cast—so it’s called “cast iron”. The low-carbon iron can be worked with tools, hence “wrought”.
It’s the smelting process that results in the carbon content: smelting at temperatures high enough to melt the iron, also causes it to undergo a phase change that causes it to absorb more carbon.
It’s more the other way around: Iron with more than ~2.1% carbon is brittle, and therefore it cannot be worked with tools; it can only be cast—so it’s called “cast iron”. The low-carbon iron can be worked with tools, hence “wrought”.
It’s the smelting process that results in the carbon content: smelting at temperatures high enough to melt the iron, also causes it to undergo a phase change that causes it to absorb more carbon.