Do you mean you would just have that one damaged side blow up? If so, that’s what I was thinking too. And if you went with that idea I mentioned about changing the plan presentation in ANH to suggest it is targeting a single reactor, it could help imply that reactor was on that side of the superstructure, explaining why only that side took most of the damage, but it would still be enough to cripple the laser system itself.
With ESB, I was thinking you change a word in the opening crawl regarding the Death Star to “devastated” rather than “destroyed”. And then in the film itself, when we see Darth Vader on the bridge of the Executor for the first time, we could see the Death Star through the bridge window, maybe less complete than it is in ROTJ, and with little specks of ships flying around it. It is basically a Death Star cameo, but this helps set up its return in ROTJ, and helps makes its return feel more intentional rather than be the result of a lack of originality or something. People will now expect for it to be a threat in the last movie. But that part is optional.
And then with ROTJ, like you said, reword the crawl to either refer to repairing the Death Star or the super laser in particular, at least very least removing any reference to it being a new Death Star. And like I mentioned above, cut a word or two from the rebel briefing. That should cover everything I think.