Part of any human endeavor, including building scientific knowledge or running a magazine with a website, is the potential for messing up.
Humans make mistakes.
Some of them are the result of deliberate choices to violate a norm. Some of them are the result of honest misunderstandings, or of misjudgments about how much control we have over conditions or events. Some of them come about in instances where we didn't really want the bad thing that happened to happen, but we didn't take the steps we reasonably could have taken to avoid that outcome, either. Sometimes we don't recognize that what we did (or neglected to do) was a mistake until we appreciate the negative impact it has.
Human fallibility seems like the kind of thing we're not going to be able to engineer out of the organism, but we probably can do better at recognizing situations where we're likely to make mistakes, at exercising more care in those conditions, and at addressing our mistakes once we've made them.
Ethically speaking, mistakes are a problem because they cause harm, or because they result from a lapse in an obligation we ought to be honoring, or both. Thus, an ethical response to messing up ought to involving addressing that harm and/or getting back on track with the obligation we fell down on. What does this look like?
1. Acknowledge the harm. This needs to be the very first thing you do. To admit you messed up, you have to recognize the mess, with no qualifications. There it is.
2. Acknowledge the experiential report of the people you have harmed. If you're serious about sharing a world (which is what ethics is all about), you need to take seriously what the people with whom your sharing that world tell you about how they feel. They have privileged access to their own lived experiences; you need to rely on their testimony of those lived experiences.
Swallow your impulse to say, "I wouldn't feel that way," or "I wouldn't have made such a big deal of that if it happened to me." Swallow as well any impulse to mount an argument from first principles about how the people telling you they were harmed should feel (especially if it's an argument that they shouldn't feel hurt at all). These arguments don't change how people actually feel -- except, perhaps, to make them feel worse because you don't seem to take the actual harm to them seriously! (See "secondary trauma".)
3. Acknowledge how what you did contributed to the harm. Spell it out without excuses. Note how your action, or your failure to act, helped bring about the bad outcome. Identify the way your action, or your failure to act, fell short of you living up to your obligations (and be clear about what you understand those obligations to be).
Undoubtedly, there will be other causal factors you can point to that also contributed to bringing about the bad outcome. Pointing them out right now will give the impression that you are dodging your responsibility. Don't do that.
4. Say you are sorry for causing the harm/falling down on the duty. Actually, you can do this earlier in the process, but doing it again won't hurt.
What will hurt is "I'm sorry if you were offended/if you were hurt" and similar locutions, since these suggest that you don't take seriously the experiential reports of the people to whom you're apologizing. (See #2 above.) If it looks like you're denying that there really was harm (or that the harm was significant), it may also look like you're not actually apologizing.
5. Identify steps you will take to avoid repeating this kind of mistake. This is closely connected to your post-mortem of what you did wrong this time (see #3 above). How are you going to change the circumstances, be more attentive to your duties, be more aware of the potential bad consequences that you didn't foresee this time? Spell out the plan.
6. Identify steps you will take to address the harm of your mistake. Sometimes a sincere apology and a clear plan for not messing up in that way again is enough. Sometimes offsetting the harm and rebuilding trust will take more.
This is another good juncture at which to listen to the people telling you they were harmed. What do they want to help mitigate that harm? What are they telling you might help them trust you again?
7. Don't demand forgiveness. Some harms hurt for a long time. Trust takes longer to establish than to destroy, and rebuilding it can take longer than it took to build the initial trust. This is a good reason to be on guard against mistakes!
8. If you get off to a bad start, admit it and stop digging. People make mistakes trying to address their mistakes. People give excuses when they should instead acknowledge their culpability. People minimize the feelings of the people to whom they're trying to apologize. It happens, but it adds an additional layer of mistakes that you ought to address.
Catch yourself. Say, "OK, I was giving an excuse, but I should just tell you that what I did was wrong, and I'm sorry it hurt you." Or, "That reason I gave you was me being defensive, and right now it's your feelings I need to prioritize." Or, "I didn't notice before that the way I was treating you was unfair. I see now that it was, and I'm going to work hard not to treat you that way again."
Addressing a mistake is not like winning an argument. In fact, it's the opposite of that: It's identifying a way that what you did wasn't successful, or defensible, or good. But this is something we have to get good at, whether we're trying to build reliable scientific knowledge or just to share a world with others.
I'm happy to entertain discussion of this particular case in the comments provided it keeps pretty close to the question of our ethical duties in explaining and apologizing. Claims about people's intent when no clear statement of that intent has been made are out-of-bounds here (although there are plenty of online spaces where you can discuss such things if you like). So are claims about legalities (since what's legal is not strictly congruent with what's ethical).
Also, if you haven't already, you should read Kate Clancy's detailed analysis of what SciAm did well and what SciAm did poorly in responding to the situation about which DNLee was blogging and in responding to the online outcry when SciAm removed her post.
Also relevant: Melanie Tannenbaum's excellent post on why we focus on intent when we should focus on impact.