but isn’t “cast” steel a reference to its carbon content?

In the process of melting ore in a crucible, or casting, one may control the aspect of carbon content as well as other constituents. I am not sure if "cast" actually refers to the end product's carbon content. The advent of crucible steel did afford better control over the hardness of steel by allowing the addition of carbon to the melt. There was a cost, however, in that phosphorus and sulfur is retained at a higher percentage than in, say, bloom steel.