You're not my SO, are ya? You're a random on the internet that is clearly not reading what I say very carefully, because (ironically) your accusations require reading more into what I said than what I actually said. No, how I communicate with some random on the internet is not how I communicate with my SO. We spent no small amount of time working on our communication and how to communicate effectively with each other. We know the ins and outs of how to communicate with each other. If you can communicate with a random over the internet just as well as your SO... well, that's either mad props to you or kind of a sad statement on communication in your relationships, and the second seems more likely than the first. I'm a bit closer to my SO than randoms on the internet.

I never said the argument was between you and the other person or the two of you united against a common problem.

No, and I never said you did, implied you did, or made an argument that required your saying it. I DID comment on your argument style, because you did say that...

I'd personally prefer a jokey "you're pretty" that gets us both past than you just telling me that we need to do this later so you can look like the peace maker while I'm high and dry and still pissed.

You explicitly stated how you would see it as the person doing it "so you can look like the peace maker." You didn't say "I view controversy as adversarial". You just said "I would handle this situation like <adversarial controversy>" You explained how you see the sides. The point is simple, if that's how you're seeing things, it basically requires an adversarial position. If you were approaching it as two people against a problem, you wouldn't see it as them trying to look good. You'd see it as them trying to help THE TWO OF YOU solve the problem more effectively. You are clearly defensive there. You don't have to say it. People very very rarely outright state "Yes, I am being defensive."

The other option is you were just presenting a false dilemma intentionally, as though the only options are joking and trying to look better than the other. I chose to give you the benefit of the doubt and assume you were arguing in good faith. If you weren't, my bad, but I have no interest in debating people who are not arguing in good faith.

If the communication in your relationship is such that you are offended because they tried to resolve a situation peacefully, I don't think that says anything remotely good about your communication. That you would even THINK to see it as "oh they're just trying to make themselves look good" reveals how you approach the situation. You're more worried about how you or the other looks than resolving the situation. To be threatened by them trying to resolve it requires viewing it as though they're an adversary, not someone trying to work with you. It's not like you can't communicate back if you're unhappy with their proposed solution. However, you can't communicate back about the solution if they just blew you off and said you're pretty, because they didn't actually work with you to propose a solution.

TLDR: Communicating is almost always better than avoiding communication in a relationship. Seeing an attempt to communicate and resolve the issue as such a problem reveals plenty about how you handle arguments.