When people don't like your point of view but they don't know how to respond to what you just said, this is what they usually do: they respond to something other than what you said. They alter what you said until they can make you sound like a stupid or bad person. They make it easier for themselves and harder for you - unfairly, I might add. Probably most of the time they don't even realize they're doing this. And their lack of awareness of that makes it quite difficult to continue discussing - or even to get them to see the bad conversational moves they're making.

A common example of this is the Straw Man argument/fallacy, but I didn't call it that because I don't think people know they're doing it, and because it isn't always an argument. Sometimes it's literally just acting as if what was said was something else, and continuing on that basis.

This larger category might be called "convenient misunderstanding." And sometimes it's really quite proud, too. People are proud to misunderstand.