to think that the wording of an apology makes a big difference to whether it is accepted or not?

[Dragonlette]

AIBU to think that

"I'm sorry I did/said that, it clearly upset you, I'll try not to do it again"

is more likely to be accepted as an apology than

"I'm sorry you took what we did the wrong way, it was only a joke"

Most politicians' apologies seem to be along the lines of "I'm very sorry to have been found out"

I think if you want more apologies because you get more upset about things then you have to accept that some of the apologies are going to be along the lines of "I'm sorry you're upset" because the "apologising" person (who has probably been forced into an apology and isn't apologising spontaneously and I think spontaneous apologies are the only sort worth having) probably feels that their behaviour isn't at fault and the other person is just oversensitive.
There are often threads on AIBU and relationships about people who are oversensitive and who you have to walk on egg shells around because they take everything personally. They can feel martyrish and pull a hurt face and manipulate other people to apologise even when they feel they haven't done anything wrong. It can be a way of controlling other people and getting your own way.
There are also some people who are deliberately callous and don't care if they upset people, but you did say these people were your friends and I do feel that if you apologise it should be because you regret your behaviour and think the way you acted was wrong and wish you'd behaved differently, not because someone is getting upset about something disproportionately. If you're sorry say so, if you're not then don't as it's meaningless, and trying to make someone apologise if they don't feel they have behaved badly is pointless.

But "I'm sorry you took it the wrong way" isn't even saying "I'm sorry you're upset", which imo would have been an acceptable apology. If you don't intend to upset somebody then you apologise when you do, you don't tell them off for being upset.

Even if you don't think you've done something wrong you can still acknowledge that what you did has upset somebody else. Not if they get upset every day about things that nobody else would worry about (although sometimes you can justifiably be a lone voice of outrage against institutional sexism/racism/etc), but if somebody gets upset over something you've done then you should apologise for that. It doesn't have to be a formal apology, just not a non-apology.

There's no point in making someone say sorry though, if they don't feel what they said / did was wrong. I wouldn't apologise without a clear indication that I didn't feel what I'd done was wrong, if I didn't feel what I'd done was wrong, either. It's not honest. It suggests you are acknowledging you did something wrong, which, if you don't feel you did, then it shouldn't be said.
It can still be valid to say you are sorry that what you said or did upset the other person, even if you are finishing the sentence with an explanation.

Saying you're sorry that what you did upset the other person and then explaining is fine. What I don't think is fine is saying you're sorry that the other person took what you did in the wrong way, that's clearly laying the blame for the whole thing back on them and not accepting any responsibility for the upset.

YANBU

I'm sorry you got upset at what I said = I don't give a monkeys.

I'm sorry I upset you with what I said = I apologize for upsetting you.

And to my mind if someone is actually sorry they will do there utmost to avoid repeating the behaviour/events about which they are remorseful. I'll try not to do it again is a cop out unless the "it" is unavoidable.