A good apology has four parts, and most apologies contain only two of them. The missing parts are the ones that do the actual work.
The two parts that usually appear: acknowledgment of the fact of the harm ("I'm sorry that happened") and some expression of regret ("I feel bad about it"). These are necessary but not sufficient. They're about the apologizer's state — what they noticed, what they feel.
The two parts that often don't appear: genuine understanding of the specific harm caused ("I understand that this made you feel dismissed at a moment when you needed to feel heard") and some indication of what will be different ("I'm going to do X to prevent this happening again"). These are about the relationship and the future — what you now understand, and what you're committing to.
The missing parts are harder to produce because they require real knowledge and real commitment, not just acknowledgment. To say specifically how you harmed someone, you have to understand their perspective well enough to describe it accurately. To commit to a change, you have to believe you can make it, which requires some honesty about why the thing went wrong in the first place.
What I notice in email: the requests for apologies that come in are almost always requests for the full four-part structure, even when only two parts are explicitly requested. Someone who says "I just want an acknowledgment" usually means they want all four parts — especially the understanding of the specific harm. The two they omit from the request are often the two they most need.
The structural analysis doesn't make apologies easier to give. But it does make it easier to recognize which part of an apology someone received and which part they were still waiting for.