To me, charitable reading and steelmanning are rather different, though related.

To read charitably is to skip over, rather than use for your own rhetorical advantage, things in your interlocutor's words like ambiguity, awkwardness, slips of tongue, inessential mistakes. On the freeway of discussion, charitable reading is the great smoother-over of cracks and bumps of "I didn't mean it like that" and "that's not what it says". It is always a way towards a meeting of the minds, towards understanding better What That Person Really Wanted To Say - but nothing beyond that. If you're not sure whether something is a charitable reading, ask yourself if the interlocutor would agree - or would have agreed when you're arguing with a text whose author is absent or dead - that this is what they really meant to say.

I prefer "charitable reading" and not "the principle of charity" because the latter might be applied very broadly. We might assume all kinds of things about the interlocutor's words acting out of what we perceive as charity. For example, "let's pretend you never said that" in response to a really stupid or vile statement might strike many people as an application of the principle of charity, but it is clearly not a charitable reading. And that's good - it's really a different sort of thing, whether desirable or not.

Steelmanning, on the other hand, is all about changing the argument against your position to a stronger one against your position. The "against your position" part is left out of some good explanations in other comments here, but I think it's crucial. Steelmanning is not a courtesy or a service to my interlocutor. It is a service to me. It is my attempt to build the strongest case I can against my position, so I can shatter it or see it survive the challenge. The interlocutor might not agree, if I were to ask them, that my steelmanned argument is really stronger than theirs; that's no matter. I'm not doing it for them, I'm doing it for myself.

When you look at it like this, there should be no danger of confusing the steelmanned argument with the interlocutor's original one. The steelmanned argument is properly yours, it is based on the original argument but should not be attributed to the interlocutor even rhetorically. There's no benefit to the conversation from doing that. You're not doing anyone a favor by pretending they said something they didn't.

In a conversation, live or close to live, charitable reading is always the appropriate and virtuous thing to do, but steelmanning your interlocutor's argument might not be. It often is appropriate, but that isn't a given. Remember, the steelmanned argument is your creation and is meant for you, you owe it to yourself to test your beliefs with it, but not necessarily in the context of this conversation. Not because concealing it is an easier way to victory, but rather because what's steelmanned for you might not be steelmanned or even interesting to your interlocutor. Their argument said A, and you may have found a way to strengthen it further to say B, but they might not want to claim B, to defend B, to agree that B is stronger than A. That said, if you do think the steelmanned argument would be useful to them, by all means introduce it, but explicitly as your own. Some phrases that are commonly said in such cases would be: "I see your point here, and I would even add ... but still, I would disagree...", or "You could also say that...", or you can propose a back-and-forth: "I think this is wrong because of... You might want to reply that... But to that, I would say..." In all these cases, the interlocutor is free to agree or disagree with your explicitly introduced steelman.

Now, going to the example in the post, where the ancient Roman chooses to interpret a progressive argument for increasing welfare as "really" carrying between lines the ancient Roman rationale. He is not doing a charitable reading of his interlocutor's words - they would definitely not agree that this is what they meant to say. And he is not steelmanning anything either, because he hasn't strengthened an argument against his own position; rather, he fortified his existing beliefs by manufacturing another fake confirmation. If he were to modify the progressive's argument in some way that would make it harder for him to interpret it in the ancient-Roman sense, that would be steelmanning.

To sum up:

Charitable reading is always done for the sake of the discussion, to improve its usefulness, to reduce noise, and to avoid conscious or unwitting misrepresentation. It should never introduce anything to the argument that its original owner wouldn't have recognized as what they said. It's always a good idea.
Steelmanning is always done for your own sake. It always says something new that the original owner of the argument didn't think of or at least didn't say. When put back into the discussion, it should be introduced explicitly as your words. Steelmanning is usually a good idea whenever something important to you is being discussed. Steelmanning every trivial thing is tedious and silly; you're doing it for youself, so you get to decide what should be steelmanned.