As someone who has made a lot of mistakes and who tries hard to admit them and apologize when I think apologies are in order, I have become somewhat of a student of the apology. I think it's important, if you apologize, to do so sincerely and actually to admit your mistake.
"Well, of course," you might say, "what other kind of apology is there?" There are two others that I know of. In my lexicon, I call them the "Jesse Jackson" apology and the "I forgot who did it" apology.
First, the Jesse Jackson apology. In January 1984, Jesse Jackson, running for the Democratic nomination for President, referred to Jews as "Hymies" and New York city as "Hymietown." When an article in the Washington Post referenced his statements, Jackson, after first lying, saying he hadn't made these statements, ended up, according to Larry Sabato, making "an emotional speech admitting guilt and seeking atonement before national Jewish leaders in a Manchester, New Hampshire synagogue." I have insufficient reason to think it wasn't an actual apology.
But then, on a much bigger stage, the nationally televised Democratic convention in San Francisco, Jackson gave what I now call the "Jesse Jackson" apology. He said (at about the 6:20 point):
If, in my low moments, in word, deed or attitude, through some error of temper, taste, or tone, I have caused anyone discomfort, created pain, or revived someone's fears, that was not my truest self.
Why the "if?" There's no if about it. With his earlier comment, he did cause people pain.
The other apology, the "I forgot who did it apology," is so common that I'm sure you can find your own examples. The key is the passive voice. That allows the person not to identify who did the thing he is apologizing for. Some standard versions are "mistakes were made" and "things were said." Someone not paying close attention would wonder who made the mistakes and who said the things. And that's the point.
I don't want to get into whether an apology was justified in each case. Instead, I want to point out how beautifully clean both apologies were.
Here's Stewart (at about the 1:35 point), responding to a district attorney who claimed that Stewart had his facts wrong:
You were right about that. District Attorney Ramos is right. We were wrong. In our list of unarmed black men shot by police, we should not have included Dante Parker, who, according to the county medical examiner, died of a PCP overdose. So I'm sorry about that. Shouldn't have done that. AHHH! I [bleep] hate making unforced errors like this. I hate it. I get so mad at myself. Stupid, stupid, stupid [while hitting his forehead.]
Many people have seen my emails with Ran Duan of Sichuan Garden restaurant in Brookline. Having reflected on my interaction with Ran, including what I said and how I said it, it's clear that I was very much out of line. I aspire to act with great respect and humility in dealing with others, no matter what the situation. Clearly I failed to do so. I am sorry, and I intend to do better in the future. I have reached out to Ran and will apologize to him personally as well.