I'm pretty sure that the iron that is stronger than Bronze is technically an early Steel.
I think you're correct, but there's the issue of purity. Or, maybe, of just what "early Steel" means.
If you bought a steel item today you'd usually expect it to be pure steel. But IIRC roman swords, for example, generally only had layers of steel from the charcoal used to heat the iron. The presence of steel was a happy accident caused by the method of manufacture - the charcoal used to heat the iron case-hardened the outer layer, and that layer got folded into the sword when the weapon was made. So the swords weren't pure steel, or even mostly steel, perhaps. But the weren't merely iron either.
So I'd say there were still "iron weapons"... but the distinction is fairly arbitrary. The
important thing, though is indeed the steel content. Too little and you may be better off with bronze. I think it'd depend on the hardening - or lack of hardening - the smith used. I've read that some iron swords had surfaces that are best considered high-carbon steel... but most didn't.
I like phrase "Iron weapons" more than "Steel weapons" for FFH. I think "Iron" has more of a fantasy-feel. OTOH, it might be nice to give the Khzad, or someone else, some steel.
Still, there's a world of difference between iron and steel.
Well.. between homogeneous steel armor and simple iron, yes. But it's more accurate to say that what's between them is a continuum. And while a lot of the early "iron" weapons were technically steel, or incorporated steel, it's still useful to speak of them as "iron". It emphasizes the differences between, say, a post-Bessemer "steel" sword and an Classical "iron" spearhead. It's just quicker than "early industrial steel" or "carburized ingot steel".
I guess, basically, I think it's a mistake to get too hung up on the word "Iron." Think "copper-alloy" and "iron-alloy".
If there were four weapon tiers then, I agree, your "bronze, iron, steel, mithril" would make good sense. But, at 3, I like "iron" better than "steel" for the reason given above.