It might be worthwhile to step back and first ask why we bother debating other people at all? Does it provide us any value?
There are at least two reasons for arguing with others:
First, we want to understand things better. Talking to others forces us to question assumptions we may not have realized we had. In the case of the flat earth, we assume the earth is not flat, but we may not be able to give good reasons why it is not flat or there may be reasons why it is not flat that we haven’t even thought about. So, debating this increases our knowledge even though we already know the earth is not flat.
Second, we want to be right about our commitments. Being social, we want others to acknowledge that we are right. We make gut decisions to accept various positions. Our commitment to these positions encourages us to justify or rationalize them. That may seem to belittle our rational abilities, but actually it enhances them. We don’t reason like some indifferent machine running through the possibilities offered by our programming. We are motivated reasoners. And since we are social we test our reasoning on other people who may not agree with us. This leads to rapid progress in knowledge. See Antonio Damasio’s Descartes’ Error: Emotion, Reason and the Human Brain for a neuroscience perspective on this.
So the reason we bother debating at all is because we are motivated to defend our positions and we want to better understand the positions we maintain.
Now, for the question: “Is there a term for his behaviour?” It could be called debating.
We should not expect other people to agree with us--ever. And they should not expect us to agree with them either. What we get out of the debate is a better understanding of our own positions. All this motivated reasoning especially in front of someone else who offers resistance to what we take for granted may lead to changes that improve our own lives if not the world around us.
This makes little difference when it comes to something like the flat earth. Few people care even if it were flat. So what if someone thinks the earth is a pancake? However, there are things some people do care about.like who will be the next president? Or do we have free will? Or how do we set up a colony of machines to explore mars for us? That we care enough about something to debate it or at least talk about it helps us get things done.