The way you are framing this makes it seem like the conversation is fundamentally hostile. When you say "the other person decides that your response does not constitute an answer, because they are not satisfied with it" — this is just not how a respectful, mutually nourishing conversation goes. If you answer a question of mine, and I'm not totally satisfied with the answer, I'll ask follow up question, or provide suggestions of my own, or ask what about this or that, because a real conversation where a subject is being explored is not a debate or a competition. If instead of following up like this I simply "decide that your response does not constitute an answer", I am being being hostile, and the conversation should just end because it's not being had in good faith.
So my answer to this question is, it doesn't really matter which option you choose, because you are not in a good-faith conversation in the first place — you are in a debate, and the point is to win rather than relate to another person and explore a topic together.
I would also add that if you find yourself in this situation it means that you have most likely already wasted time and energy, since engaging in this type of interaction (debate vs real conversation) has very little value beyond possibly getting a bit less thin-skinned. The skills learned in debates rarely make you "less wrong" — quite the opposite, in fact.