Don't give what's yours, and don't take what's theirs.
Here's what I mean by that statement.
You are responsible for your words, actions, perceptions, and feelings. Don't give that responsibility to them.
They are responsible for their words, actions, perceptions, and feelings. Don't take that responsibility from them.
If you say something like “I'm sorry that I made you feel that way,” you have taken responsibility for their feelings. I strongly recommend that you apologize for your words and actions that contributed to the conflict. I also recommend that you NOT take responsibility for their feelings.
If you say something like “You made me angry,” you have given responsibility for your feelings to the other person. They are responsible for their words and actions. They are not responsible for your feelings.
Using a “When you _____, I felt _____” type of statement avoids the problems of giving what's yours and taking what's theirs. This approach communicates in a stronger, clearer, and more accurate way the reality of the situation. They exhibited a specific behavior, for which they are responsible, and you perceived their behavior in a way that created a specific emotion, for which you are responsible.
Likewise, “I'm sorry that I said (or did) ____” keeps your responsibility clear. You are responsible for your actions, not their feelings.
Clearly defining responsibility for different portions of an interpersonal conflict reduces the odds of conflict escalation and improves the odds of effective conflict resolution.