This question was asked a long time ago, but never received an answer. I largely agree with Deane Yang's comment, but wish to expand on it, for the sake of future visitors to MO.
First, several other questions have been asked regarding how to deal with errors in the published literature:
Diplomacy when reporting errors
How do I fix someone's published error?
Refereeing a paper containing strong statements about other papers
As an author, if you find a mistake in someone else's paper, you should be guided by a combination of ethics, humanity, and what's best for your readers. Remember that the authors of that other paper are human, and almost certainly did not intend to make a mistake. Making a mistake is a nightmare scenario to most mathematicians, so please keep in mind that they are probably embarrassed and might instinctively react in a defensive way.
The first step after finding a mistake is to check your work with someone else, and be sure you can explain exactly where the mistake occurs. Even better if you can produce a counterexample to the wrong statement, or a correction to the proof (possibly by changing the hypotheses and/or conclusion of the statement). Next, you should contact the authors. It's best to do this diplomatically, guided by the advice above. As a professional courtesy, it's best if they can learn about their mistake from you in a personalized email, rather than from a new preprint on arXiv. If you are very junior, your advisor can also help here. In the case of the OP, the authors already knew of their mistake, and had already written text showing that the main conclusion was still true, even if the proof had a flaw. Still, I would write to them to let them know I'd be drawing attention to that proof, but I'd frame it positively, by pointing out that I was building on their work and had figured out a way to get some mileage out of their original proof. This thread might be relevant:
Contacting an eminent mathematician
In terms of writing your paper, it's always best to think about what's best for your reader. In this case, clearly the reader needs to have a reference to the "partially wrong" version because that's where the wrong proof is written, which you build on. It's also clear that you should cite the arXiv version where the error is acknowledged. This is important for the reader to not be misled (since the "partially wrong" paper lacks a published erratum) and also for the sake of your referee, to point out that the authors have already acknowledged their error, so the referee doesn't need to check that part of your claims. Lastly, if you think the current state of the literature is unclear, this is your chance to clarify it, e.g., by explaining exactly what's wrong in the wrong proof, why the special case is still true (if this is not explained in the latest arXiv version), why that's sufficient for the original claimed result (if that's not explained in the latest arXiv version), and what if anything all this has to do with your result. Cleaning up the literature is a service for everyone.
The OP didn't ask for examples, but I'll give some anyway, because it's actually fairly common to come across papers with errors. As discussed in this answer, one of my earliest papers had to contend with an erroneous claim. It was important to do so because my paper could be viewed in some sense as correcting that error. Fortunately, the author had already acknowledged it (on MathOverflow, and I cited the relevant thread). Another situation happened not long after, when I needed a result from someone else's paper which they had recently written a corrigendum for. I cited both the published paper and corrigendum. I know of another case where people cite both the published paper and the correction. Lastly, I just read a case today, posted to arXiv yesterday (Appendix D), that does a great job spelling out why a cited proof is not quite correct, but how to fix it up.
