How to publish a corrected proof for a claimed mathematical result? I work in the field of theoretical computer science, and I have published a paper with several results. A few years later, a reader reported to us an error in the paper. The paper claims a theorem X, but the proof given actually shows a somewhat weaker result X'.
Thinking about it, I have been able to come up with a proof of the originally claimed result X, using somewhat different techniques. The new proof is not interesting to merit publication as a new paper, but the fix is not trivial either so I'm not fine with leaving the mistake as-is. Besides, result X is not central in the paper, and it is indeed correct (only the proof was not), so this is not a situation where we should think about retracting the original paper altogether.
I'd be interested in the community's opinion about what's the best way to proceed to publish this fix to an erroneous proof. I can see at least two options:

Option 1: publish a revised version of the paper with the original claim of X and the new proof (and a comment that the original version of the paper had a mistake). I think this would be the most comfortable option for future readers. That said, it may create confusion as the original buggy version of the paper will still be around. Plus, I would imagine that ethically I cannot do this without approval from all authors of the original paper (at least one of which has now left academia).
Option 2: publish a separate note on arXiv containing just the new proof of X, and publish a revised version of the paper providing the proof of X' plus a pointer to the new note for the proof of X. This has the advantage of leaving the original material essentially intact, plus the note can be authored only by the people who are still involved. But it is less pleasant to the reader (who must jump to ta different document), and it is more work for us (since one would need to re-explain the whole context in the note, etc.), all for a not-so-interesting fix.

[With both options, I would publish the updated version of the paper simply on arXiv -- dealing with the publisher would only complicate things I think.]
What do you think would be the best option? Essentially the question can be summed up as: when is it better to fix errors in a paper simply by updating it, and when is it better to publish the fix as a separate note?
I had a similar problem with one of my conference papers. We posted a revised version on arxiv, specifying in a footnote what the mistake was and how it was corrected. The correct proof made it official appearance in a journal paper that was partially based on the the conference one.