An incorrect moderator action was taken by someone else, and I am incorrectly being blamed by other users for taking these actions. How would I go about resolving this with the community peacefully?
I agree with cw1998's post about making an announcement, but I think this is more an issue of balancing user privacy and your happiness, and you should be very careful about how you handle it with these specific regards:
Don't give out information that the system doesn't presently give them.
If the community can't tell which moderator took the action, which reversed it, or which user had been incorrectly sanctioned, don't announce it. You may not like being the focus of people's attention, but it's preferable to the user (who was innocent in this case) being brought to public attention or a public statement which amounts to one mod calling out another, which would be very bad for community cohesion (and your relationship with the other moderator).
Tell the community how you are going to prevent this from happening again.
For example, create a moderator-only announcement to remind them of the rules in this regard.
Deal with it.
Being a highly visible member of the community means you will always have to deal with negative reactions to decisions it appears you (or your fellow moderators/administrators) made. If their communication with you doesn't break the rules, just tell them the situation has been dealt with and leave it at that.
If you have the permissions, I would create an announcement explaining the entire situation in full. Preferably I would get an admin to post such an announcement as, to be honest, not as many people will be against the admin as they would maybe a moderator. And also the admin will probably have a stronger voice in the community and more people will take his or her side.
The first step is to figure out if it was an incorrect action or simply one you personally disagree with. If you are in a position to over-rule a moderator without them feeling slighted, then this doesn't really matter. If you are not, then discuss the action with the applicable moderator and reach a conclusion about the action in a way that won't cause problems amongst the moderation team itself.
If it is deemed that the action was in error, simply indicate that you disagreed with the action, you were not the one responsible, but you wish to protect the privacy of the person who is (unless it was severe enough of an issue to merit visible disciplinary action against the mod), and then reverse the action. Some users may still believe it was originally you, but if you aren't standing behind the action, most people will take you at your word. If you talked with the moderator and they also agreed with the reversal, it is probably worth mentioning that the original moderator is also on-board with the change.
If you are not able to come to an agreement without causing further issues within the moderator community, it may be best to let it be as a correct action you disagree with. You may want to consider getting another moderator to confirm you were not behind the decision (this is fairly safe to do as long as it doesn't give away who was actually responsible) or even to request that the actual moderator take responsibility for their action so that you don't have to take flack from the community on something you are in disagreement about.
A moderator team is responsible for actions that a member of that team makes. It doesn't really matter whether you personally took the action or another moderator did.
You have basically three choices:
You talk with your moderator team and decide that the action was wrong. If you come to that conclusion you excuse yourself to the user and promise to act differently in the future.
You go and defend the action. You say that it violates the forum policy and explain which part of the forum policy got violated. Whether or not you personally agree with the action doesn't matter. Internal disagreements of a moderation team shouldn't be fought in the open. Being a team means defending your fellow moderators. Depending on how your forum works you might also be very sparse with explaining the issue.
You ignore the issue and let another moderator deal with it.