I still act as if the thing is not true. Then, by some mysterious process, the thing will "click" and it feels like system 1 really gets it. After this the belief in the thing is reflected in behavior.

The "mysterious process" is the translation from an abstract concept to a specific experience (or set of experiences), either real or imaginary. That's why "fictional evidence" influences people's behavior more than abstract discussion, and why simulation games are better still.