The fallacy is thinking that we must predict an exact state vs. a probabilistic prediction of an information-theoretically meaningful set of states. Reality is metastable, so there are certain attractor states, even if randomness is injected throughout.
I have no idea the exact second my plane is going to land tomorrow, but I’m reasonably confident it will land, and I’m very confident that the United States will still exist tomorrow.
I have no idea if the United States will exist in 10,000 years, but I’m pretty sure Earth will still be a rock orbiting the Sun at that time.
And so on.
Using your analogy, whether United States will exist in 10000 years might actually be undetermined. Not just unknown, unpredicted, but undetermined in the chaos theory- sense. Not only you don't know it - the universe itself does not know it. And the universe is built in such a way that it is impossible to know. Not just "very hard and requires a lot of computation". Impossible, because we can show that the computations you would need would require a computer bigger than the universe itself.
Then there is of course an extra note required here is that there is no way for me as a human to know if the question about United States actually is universally undetermined - or just unpredicted; if there is some entity that can predict that. But the chaos theory says that many aspects of the system, and the system as a whole is undetermined. (While having exceptions where simple things like your plane landing can be predicted with some accuracy, yes.)
Also please note, and I repeat myself that I argue there is a great difference between "really hard to predict" and "physically impossible to predict for any being, based on our own notion of the universe".
If it "means nothing in practice", and can be true/not true regardless of whether it is possible (even in theory) to test it - then I assume that it is unrelated to science and you are using it in a philosophical sense?
Turing machines are deterministic. Enumerating all Turing machines is deterministic. Whether any given Turing machine will terminate is unpredictable (the Halting problem).
Unpredictability does not entail nondeterminism, although distinguishing the two is not necessarily always possible.
Paraphrasing, the theoretical image of an atom, as well as the set of atoms and other particles - is perfectly deterministic. But the chaos theory talks about the real world, not the theoretical framework.
Chaos theory is a very challenging and interesting concept which you should check out, but one of the most mundane outcomes of it is that there is no way to predict the physical world. No way to see what is going to happen. Now, obviously one could predict a simple neutonian mechanical system for a 10 second-period, or predict enough of electrical impulses in a piece of designed silicon to make a functional processor - but in general, in the widest sense, there is no way to predict the universe. Even with the assumption that there is some completely determined process at the microscopic core of every interaction - those processes, when combined into big systems, wired up with feedback loops and put in a system so tiny that one has to had a machine 2 times bigger than the original system in the first place: and we arrive at total unpredictability of the universe.
Note, this is not just practical unpredictability: the question is not about us not having enough technical prowess to predict the universe. The issue is that it is physically impossible to build any real prediction machine that would predic the whole thing: you would need a bigger universe than the universe it is trying to predict. It is even a logical contradiction!
Thus determinism becomes more of a philosophical outlook than an actual physical property that could be calculated, predicted etc.
Even the word itself - "determinism". Assumnig that the fate, the outcome of something is determined. In what way can it be determined if we know that it is physically impossible to determine it? Not because we lack machines or havent built a strong enough microscope. Our very own scientific paradigm, the scientific method, the way of thinking about the universe - the thing that lies at the core of our worldview - says that it is impossible to predict the whole system. Absolutely impossible. So it is unpredictable - ergo undetermined.