What is owed afterwards

Two separate accounts, and people often settle one and forget the other.

There is what is owed to Allah, which is repentance, and there is what is owed to the person, which is not the same thing and is not covered by the first. Words said in anger land on somebody, and an insult does not stop existing because you regret it privately.

That distinction is the tradition's own and not ours. Sahih al-Bukhari 6534, graded sound, opens with it: whoever has wronged his brother should seek his pardon in this life. We are quoting the first clause and stopping there on purpose. The rest of that report is an account of how the debt gets settled afterwards, and somebody who is already ashamed does not need to be handed a calculation of what he has lost.

What repair looks like in practice is short. Name the specific thing you said. Do not explain why they made you say it, because that is not an apology, it is a second round with a softer voice. Do not ask them to reassure you that it was fine.

One line needs drawing here, because it is easy to get this wrong in the other direction. This is about words that landed. Where somebody cursed a person privately and that person never heard it, there is a recorded position, on islamqa in fatwa 5733, that he repents to Allah and does not go and tell them, precisely so that he does not create a fresh injury in the name of clearing his own conscience. That is a different case from this one. Naming the specific thing you said is for the thing they heard, and repair is for them rather than for your relief.

Whether particular words carry a further consequence in the religion, and what a pronouncement made in anger amounts to, is the mufti's question from lesson three. Repair is not conditional on that answer.