Observe the flow of causality/information.

I suggest to stop treating this as a philosophical problem, and approach it as an engineering problem. (Philosophers are rewarded for generating smart-sounding sequences of words. Engineers are rewarded for designing technical solutions that work.) Suppose you have a technology of 24th century at your disposal, how exactly would you simulate "an old man walking his dog"? If it helps, imagine that this is your PhD project.

One possibility would be to get some atomic scanner, and scan some old man with his dog. If that is not possible, e.g. because the GDPR still applies in the 24th century, just download some generic model of human and canine physiology and run it -- that is, simulate how the individual atoms move according to the laws of physics. This is how you get a simulation of "an old man walking his dog".

Is it simultaneously a simulation of "a stick that magically keeps changing its size so that the size represents a binary encoding of an old man walking his dog"? Yes. But the important difference -- and the reason why I called the stick "magical" -- is that the behavior of the stick is completely unlike the usual laws of physics.

Like, if you want to compute the "old man walking his dog" at the next moment of time, you need to look at positions and momentum of all atoms, and then calculate their positions and momentum in the next fraction of second. Ignoring the computing power necessary to do this, the algorithm is kinda simple. But if you want to compute the "magical stick" at the next moment of time... the only way to do this is to decode the information stored in the current length of the stick, update the information, and then encode it again. In other words, the simulation of the old man and his dog is in some sense direct, while the simulation of the stick is effectively a simulation of the old man and his dog... afterwards transformed into the length of the stick. You are simulating a stick that "contains" the old man and his dog. Not a normal stick. (Similarly, you could simulate a magical "Monday morning Miami" that "contains" the old man and his dog, e.g. encoded as current positions of ants in the city. But you couldn't simulate a normal Monday morning Miami like that. The ants would not move in the same pattern as normal ants do.)

tl;dr -- "how exactly do you calculate the next moment of your simulation?" is the key question here

For your second question, observe in which direction of your simulated universe you can run computations. Imagine that as a god outside that universe, you are allowed to make exactly one intervention: to insert into the universe at some point a computer that will (within the universe) compute e.g. the prime numbers. Ok, so... here is the place where you inserted the computer... and where exactly do you expect that the results (the computed prime numbers) will appear? That is the in-universe direction of time.

If you inserted a computer into our universe at some place on Earth, the logical place to look for the results would be the same place on Earth, some moment later in the future. If you insert a computer into a realistic comic book, the place to look for the results is a few pages later to the right (or to the left if it is a Japanese manga).

This seems legit… I just would like to have a more precise understanding of what this ‘computation on-demand’ property is.

Well, you can believe that its some kind of physical causation without breaking physicalism.