The way this works is that the image is designed to appear 'beyond' the surface it is printed on. It's much easier to relax your eyes and pretend you're looking at what's 'behind' the paper. Kind of like 3d chalk art on the road in a way.
The other way of crossing your eyes works because you're swapping the left and right eye, which gives a different, inverted appearance. Instead of a foreground image popping out of the background, it looks like the other way. Like looking in a box, kinda.
I can do both, but the latter is more difficult, sometimes requires a specific distance, and can be painful if you force it. If the image is too big, you may only be able to see a part of it. I think the first method is easier to do and to learn/train. Either way, you aren't looking at what's 'on the surface'.
...
The best way I can explain is: pretend you're sitting on the toilet, really tired and you have nothing to look at so you just lose focus and gaze at random stuff. When the tiles or cracks start to make pictures that aren't there, that's kind of the effect you want.
Oh that one is a good one, it's very busy. Using the first method the trees are on the 'bottom' and everything progressively pops out with the fish/turtle on 'top'.
The other way is reverse, the trees are on the 'top' and the fish are on the 'bottom' (like I'm looking in that 'box'). It's also really hard to see the whole picture this way, but that's just me.
Also, 'In a Box' might not be the best analogy, you can make one that intentionally feels like you're looking inside something -- it's just that most of these are made to pop out at you.