Newtonian is just plain determinism. One thing causes another, but does not imply things are fixed in any particular way.

Superdeterminism implies that things are fixed in advance in such a way that the things we "choose" to measure always looks as if there are no local hidden variables (e.g. Bell's inequality) even though there are.

No, they are completely different. Determinism is the idea that if you have a perfect snapshot of the universe at one time, you could run that snapshot forwards or backwards using the laws of physics and have a complete history of the entire universe. It's a valid scientist idea, but Bell's Inequality experiments pretty much prove it false by showing that either realism or locality is false, and you kind of need both to have determinism.

Meanwhile, superdeterminism is the idea that you can still have realism and locality by claiming that Bell's Inequality experiments are invalid because the universe could somehow "know" you are performing a Bell's Inequality experiment and rig it to get the "wrong" answer. Superdeterminism is equivalent to saying that we can't prove smoking causes lung cancer, because the universe might rig any science experiments to make it look like smoking causes cancer. Anytime you ask why the universe would do that or how you can prove anything in such a universe, they shrug their shoulders. It's a completely BS idea that got traction because specific people pushed it and others don't understand quantum physics enough to know how unserious of an idea it is.

You can imagine the general idea by imagining the state space of a bunch of billiard balls: their position and velocity in a 6D space.

This is a metaphor and not the state or experiment that superdeterminism seeks to answer. I am also speaking well outside the areas of physics I was trained in. So huge grain of salt.

In classical determinism, the trajectories of all billard balls can be derived precisely from their initial conditions. Due to sensitivity to initial conditions and our ability to know them precisely, we can see chaotic mixing behavior where the trajectory of even two billiard balls that start at nearly the same place become more and more uncorrelated as time goes on as they bounce around and into each other. One expects that many accessible observables and functions loose detectable correlations as this billiard ball universe goes on and the billiard balls go on to roughly fill the full 6D state space up to conservation of momentum.

Now imagine the weirder scenario, that there exists some function and statistical comparison you can make between state variables from sets of billiard balls. Although these state variables themselves seem uncorrelated after enough time has passed, there are these properties that always hold with perfect correlation even after infinite chaotic time bouncing around. The points in state space that the balls cover is restricted to some manifold where these statistical properties are true when checked. The system cannot evolve in a way where those correlations of bulk quantities would disappear.

In this toy metaphor it’s not so horrifically odd. Superdetermism gets more flak because: (i) the state space and property in question is so much weirder and more complicated than the kinematic example, the correlations are between objects that measure and the objects being measured! (ii) if these properties are always true under any check we can make there is no way to falsify. Then it doesn’t matter if the behavior is from some unimaginable underlying universal state space geometry or local quantum effects. This same argument could also be applied more generally and inappropriately to reduce any experiment to “that’s just the way the structure of the universe is”.

It's quite deeply weird. It's basically looking at the seemingly impossible correlations that occur in quantum mechanics and saying that they happen because the decision about what to measure is correlated with the state of the system somehow (Bell's inequality assumes that these are independent). This is very strange in a universe where we apparently could base that decision on many different seemingly unrelated processes. (including processes that have been causally seperated for near the lifetime of the universe, e.g. by choosing the measurements based on noise in the CMB).