I agree that these terms —especially 'locality'— are used for different concepts and this is annoying. I will list several notions of causality and locality.
Causality (or Einsteinian locality): Results of experiments carried out at a space-like distance are not correlated. This assumes that there are not previous correlations before making the experiment. In a quantum theory, this implies that observables must commute at a space-like distance. A violation of this property in a Special Relativistic theory may give raise to time travels paradoxes.
Micro-causality (only used in field theory): A theory whose fundamental variables or the dynamical degrees of freedom commute at space-like distance is sometimes said to be local. Note that there are theories which violate this property without violating causality.
Lagrangian density (only used in field theory): A theory whose action functional is expressed as a space-time integral of a local Lagrangian density is sometimes said to be local. Note that there are theories which violate this property without violating causality.
Non-local character of gravity: Here I will mention a few points, which are not independent of each other:
Equivalence principle: Free-fall observers cannot detect the gravitational field by making local experiments. All they feel are (non-local) tidal forces. This implies that we cannot define a gravitational energy-momentum tensor.
Gravitational entropy is not an extensive magnitude: The entropy of a Schwarzschild black hole does not scale with the volume, but with the horizon area. This means that the degrees of freedom are not local, but live on the boundary.
Usually, as we increase energy in a theory, we explore lower and lower distances and find out substructure and new particles/degrees-of-freedom (atoms, nuclei, nucleons, quarks). However, when we take into account gravity, when we increase the energy sufficiently, we inevitably form a black hole, which are not substructure o new, more-fundamental degrees of freedom, but a solution of the theory.
Horizons: A free-fall observer feels nothing when they across the horizon (if the horizon is large enough, otherwise they would get spaghettificated due to tidal forces (non-local effect)). The space-time curvature at the horizon can be arbitrarily small. However, this is a physical boundary, in the sense of being a non-return point.
Cluster decomposition principle: Results of experiments carried out at the same time (in a given reference-frame) but in different spatial regions are not correlated. This assumes that there are not previous correlations before making the experiment. This property together with Poincare invariance implies causality. Relativistic QFT and non-relativistic condensed matter QFT verify this notion of locality. This property imposes certain smooth dependence of the Hamiltonian density on the creation/annihilation operators.
Non-local wave-function's collapse (quantum notion): People sometimes say that quantum mechanics is non-local because the collapse is a non-local process. In my opinion, the wave-function collapse is not a physical process, but something that affects our mathematical description of the physical system. A short of algorithm to incorporate new information to the theory. So, in my opinion, this non-local collapse is not a signal of physical non-locality.
Non-local states and observables (quantum notion): In quantum mechanics there exist non-localized states and observables and I have heard people to call this non-locality. A particle whose linear momentum is very well defined may be an example of a non-localized state and the scattering operator may be an example of a non-local observable given that relate states at far past and future. Entangled states could be listed here.
Entanglement (or non-local correlations, quantum notion): In quantum mechanics there are quantum correlations which are non-local such as the spin correlations of the singlet state. One needs to measure the z component of both particles which can be very separated. A singlet state may also be called non-local. Since one cannot use these correlations to send information (a parallel classical channel is required, and this cannot be superluminal), therefore this property does not imply violations of causality.
Incompatibility of QM with local (causal) realism: QM and experiments violate Bell's inequalities. This leads to the incompatibility of QM and nature with either local (causal) realism (local hidden variables) or free will. People are currently discussing this on this site.
Only the first notion of locality (causality) must be required in a Poincare invariant theory. Semantic issue: Some people call a theory "special-relativistic" if the theory is Poincare invariant and causal, while other people by "special-relativistic" just mean Poincare invariant.
- Non-relativistic quantum mechanics and non-relativistic QFT verify the cluster decomposition principle. However, there are non-local observables, entanglement, etc.
- Relativistic QFT verify causality ans the cluster decomposition principle.
- Quantum Electrodynamics in Coulomb gauge is a relativistic QFT, but it does not have a Lagrangian or Hamiltonian density.
- Para-statistics theories verify the cluster decomposition principle, but they do not verify micro-causality.
- Classical General Relativity has a local Lagrangian density and it is locally Lorentz invariant, but it allows time machines in some topologies (and in these cases is not a causal theory) and horizons have non-local properties.
I would be thankful if you correct this answer or make it more precise.