What does "causally connected" or "causes" really mean?

Further, given that we know that we can have instantaneous effects in typical quantum processes (e.g. flip the polarizer, effect on another a reading light years away, even though we cannot transmit useful information with this), does that not constitute "causing" for the purpose of this statement?

Correct, that does not constitute "causing" in this context. For the same reason I wouldn't call this sort of thing an "effect." It's really just a correlation.

In general, causality is just what you'd think it is: if event A causes event B, that means the physical state at (location,time) B has some logical/mathematical dependency on the physical state at (location,time) A, such that what happens at A influences what happens at B. Just how that dependency is expressed depends on which physical theory you're using. In classical mechanics, the dependency exists if a particle is able to travel from A to B. In quantum field theory, the dependency exists if quantum operators at A and B don't commute with each other. And so on. The exact mathematical expressions can get a bit technical if you're not familiar with the theories, but they're all expressing that same underlying idea.

The idea that there are instantaneous effects of the kind you describe is wrong. It assumes wrongly that if the outcomes of measurements are described by stochastic variables. Given that false assumption those variables have to be non-local to reproduce the correlations predicted by quantum mechanics. Quantum mechanics isn't a theory about classical stochastic variables. Rather, the physical quantities that describe the evolution of a quantum system are Hermitian operators that evolve entirely locally:

http://xxx.lanl.gov/abs/quant-ph/9906007

http://arxiv.org/abs/1109.6223

These operators describe physical reality as being a more complex structure than the universe as described by classical physics that, in some approximations, resembles multiple non-interacting versions of the world as described by classical physics.

For each measurement there will be two versions of the measuring apparatus after the measurement. One of the versions of the measuring apparatus will record spin up, the other will record spin down. When a joint measurement is done on records of each result they then become correlated.

In quantum mechanics, causation is described by the patterns of dependence among Heisenberg picture observables and these patterns do not change with changes of reference frame.