Beyond type 1 vs. type 2 processing: the tri-dimensional way (link)

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.

Re the linked article, I don't feel that the mental effort level is an independent dimension. Automatic subconscious tasks are generally perceived as "easy", to the degree that it's hard to alieve that they were once hard.

Doing math, for instance, seems to strongly have factors of both.

For most people math is certainly harder to internalize than, say, catching a ball. However, after the "click" you do not need to think much about, say, what f(x) means, the notation becomes intuitive. Similarly, you can see answers to familiar math problems without thinking about them: 2x+5=0 has exactly one solution, obviously!

Some people are significantly better at internalizing math than others (Rain Man and Good Will Hunting are classic, if fictional, examples). It does not feel like "doing math" for them, it just makes sense. Others can be trained to do it to various degrees, before in becomes a losing battle. Yet others are incapable to internalize even very simple (to you and me) math, they have to laboriously go through the steps every time, and be prodded along and corrected. But it does not look to me that there is a noticeable cluster where internalized skills still require a significant mental effort.

Are there examples in the different octants suggested by this? In particular, is there an example of something automatic, but slow and effortful?

That "click" is the sharp divide.