I'm also not sure why LCD is used as the example. Wouldn't a cellular automata (or a simplification of one) be a better example here?

The question of "time" and what you might call "metatime" are two different ones. "How does time work?" is a legitimately interesting question on its own without considering "metatime", and amenable to at least a certain amount of investigation. "Does time perhaps get scrambled around in ways we can't possibly observe?" is at least academically interesting, but in contrast, not very amenable to investigation. How could we prove that the multiple-world hypothesis is true? How can we prove that the universe does not simply stop five minutes from now? Or ram into what proves to be something else that we intersect in our future? How could we prove whether or not the Great Simulator does or does not roll back the simulation sometimes because It did not like the particular set of results? If there are ways of investigating these questions, they are much less obvious.

Warning: Naïve laymen "answer" from me :)

I think the 'direction' is the same 'direction' describing the process of say, a non-reversible chemical reaction, like catching something on fire. The reaction happens in one 'direction' only and cannot be reversed. Maybe? Of course, there are some quantum phenomena that do not obey this directionality?