When people end debates, they usually don’t even try to make an objective claim that they won (or, impersonally, that particular arguments were adequate to reach particular conclusions). Here’s a standard form for an initial impasse statement which is worth knowing:
Here is my understanding of the situation: My arguments, X, Y and Z, were adequate to objectively establish my conclusion, C. Those arguments were conclusive. I said enough that you should be persuaded. A rational, unbiased observer would agree that those arguments won the debate and would tentatively accept the conclusion C. You gave no substantive replies to X, Y or Z (except for one reply to X, which was refuted by my reply X2, which you did not substantively respond to). You made no substantive arguments about C that I didn’t address. There are currently no open questions left. So you should concede, change your mind, and start learning about C. Or else, say something new to change the situation. But you won’t do any of those, which is an impasse.
Most people end discussions with less than this. Not an abbreviated version of it (which could be fine) but less content. They don’t even claim to have won the debate. If you can’t even make a statement like this, you have no business giving up on the discussion.
To have won a debate, you have to be able to claim to have won the debate. That’s necessary but insufficient. It’s required but it’s not enough. Your claim could be wrong. But if you don’t even have a claim like that, you didn’t win. That means making a claim like “In [quote] and [quote2] I gave compelling arguments. I tried several times to get you to engage with them. You didn’t. They were adequate that they should have persuaded a reasonable person.” Making a claim like that is the bare minimum needed to think you won the debate. Actually checking whether that claim is true is a more advanced step but most people don’t even make the claim. They don’t realize that’s the sort of thing that would indicate you won a debate. They don’t know that’s the goal.
Your goal in a debate is to objectively establish some claims. Make adequate arguments for why those claims are correct. Adequate means they should be persuasive to an intelligent, objective, neutral observer. People can easily be wrong about what’s objective or adequate but they should at least try to do this. But most people, in most debates, won’t even claim to have done it and seem to be unaware they were supposed to. People quit discussions without ever saying something like “I thought my argument X was adequate and your responses were missing the point. You’re not listened, dumb or biased or something so I’m going to move on.” That’s not great but it’s better than what people commonly do.
People usually stop discussing without even any claim to have provided information adequate to change the other person’s mind or help them learn better ideas. That’s irrational.
People usually end discussions without even claiming the other person did something specific wrong, was unreasonable about a particular issue, was irrational in a way they’ve stated (rather than just irrational in general), incorrectly judged any particular argument (with reasons the judgment is incorrect), or anything like that. Pointing out a specific, impersonal flaw with the discussion (rather than the person’s ideas) would also work. If you won’t point out any problem with the discussion or with the person, you’re in no position to end the discussion in your favor. And if you have no rational claim that you won, you should consider that you lost and you’re leaving to avoid facing superior ideas.
People often make general, non-specific comments about how bad the other person is or how dumb their arguments are. These do not attempt to rationally and objectively establish any particular claims about what happened in the discussion. They don’t provide a case for why they’re right which they could honestly think was conclusive.
When debating, you should try to make an objective case for some conclusions. You should get to a point where you can claim (in your own opinion) that your arguments were objective, rational and conclusive and the debate should now be over. If you don’t do that, don’t claim you conclusively won the debate. There’s a lot more needed, but this is a basic starting point that people often don’t even reach.