Does the Bell test preclude localism, realism, both, or just one of either (indeterminate)? I saw this excerpt from the wikipedia article on EPR paradox

They postulate that these elements of reality are, in modern terminology, local, in the sense that each belongs to a certain point in spacetime. Each element may, again in modern terminology, only be influenced by events which are located in the backward light cone of its point in spacetime (i.e., the past). These claims are founded on assumptions about nature that constitute what is now known as local realism.

Local realism is made up of the notions of locality and realism. Realism being the idea that nature exists independently of the observer, while the principle of locality states that an object is directly influenced only by its immediate surrounding.
Does the test preclude both locality and realism, or all that we can say is "local realism"? That is, that both things cannot simultaneously be true.
 A: The Bell test rules out naive realism, which is what is described in the Wikipedia excerpt. That is to say it rules out the idea that that elements of reality belong to points in spacetime. Actually this was already ruled out, first by Einstein's 1905 treatment of special relativity which makes it clear that spacetime only exists as the operational results of measurement, and then by the mathematical structure of quantum mechanics, which makes it clear that spacetime cannot exist as a structure in its own right at all.
As Eddington put it (1923) “A physical quantity is defined by the series of operations and calculations of which it is the result.” IOW physical quantities have no other meaning than that generated by measurement processes determined by observers. In particular this applies to spacetime coordinates as physical quantities.
Eddington was speaking of relativity, but in the context of quantum mechanics Dirac observed

“In the general case we cannot speak of an observable having a value for a particular
state, but we can … speak of the probability of its having a specified
value for the state, meaning the probability of this specified value being obtained
when one makes a measurement of the observable.” -- Dirac P.A.M., 1958, Quantum Mechanics, p47.

The corollary is that the property of position as described by spacetime does not exist in the general case.
Bell’s theorem, taken together with the Aspect experiment, only rules out the idea of a classical explanation for Bell tests.

“In a theory in which parameters are added to quantum mechanics to
determine the results of individual measurements, without changing the
statistical predictions, there must be a mechanism whereby the setting
of one measuring device can influence the reading of another
instrument, however remote. Moreover, the signal involved must
propagate instantaneously.” — John Stewart Bell, 1964, On the Einstein
Podolsky Rosen Paradox

This is much weaker than the von Neumann theorem and the Kochen-Specker theorem, which rule out hidden variables.
It is wrong to say that Bell’s theorem (or any of the no-hidden variables theorems) rule out one of locality, causality, and realism. Nothing rules out realism, and one only has to redefine locality and causality in a more correct way, instead of taking a naive classical view of what locality and causality mean. Your own description of realism and locality is much more accurate. The following statements (from The Large and the Small) do not conflict with quantum mechanics

*

*Locality: A particle is in contact with another when it interacts
with it. A particle is within a neighbourhood of another if there is
a non-negligible probability that, in a small proper time period, a
photon can be emitted by one particle and absorbed by the other, and
then a photon can be emitted by the second particle and absorbed by
the first.

*Causality: There is a causal relation between two measurements if the outcome of one measurement alters the probability of the outcome
of the other.

Indeed the locality, or microcausality condition (which concerns the commutator of field operators) is fundamental to quantum field theories.
Two things are ruled out by quantum mechanics: determinism, and prior space or spacetime.;
There never has been a good reason to believe in determinism, which is actually over-deterministic because every time determines every other time as well as philosophically repugnant because it denies even the possibility of free will.
Feynman has given us a model, using Feynman diagrams, in which reality has a material structure which does not invoke space or spacetime at a fundamental level. Mathematically Feynman diagrams are graphs. In a graph, the configuration of lines and vertices has meaning, the paper on which it is drawn does not. Spacetime appears only as an emergent property. A full mathematical development is contained in my books and in Mathematical Implications of Relationism
A: The violation of Bell inequalities proves that quantum mechanics is incompatible with the assumption of local realism (rather than with the assumptions of locality and realism), in the sense that either the one or the other or even both turn out to be false.
