I hadn't actually found the system 1/system 2 meme before this, but it maps nicely onto how I handle those situations. The main trick is to make lots of little leaps of logic, instead of one big one, while pushing as few emotional buttons as you can get away with, and using the emotional buttons you do push to guide the conversation along.

An example of that is here. In the original example, telling someone directly that they're wrong pushes all kinds of emotional buttons, and a fully thought out explanation of why is obviously too much for them to handle with system one, so it's going to fall flat, unless they want to understand why they're wrong, which you've already interfered with by pushing their buttons.

In my example, I made a much smaller leap of logic - "you're using a different definition of 'okay' than most people do" - which can be parsed by system one, I think. I also used social signaling rather than words to communicate that the definition is not okay, which is a good idea because social signaling can communicate that with much more finesse and fewer emotional buttons pushed, and because people are simply wired to go along with that kind of influence more easily.