Bell's theorem and Bell's inequality are related to the concept of entanglement in quantum mechanics and the correlations that can be observed between entangled particles. They are not directly about the spin of particles along different axes, but I can explain how they are connected.
In the famous thought experiment known as the Einstein-Podolsky-Rosen (EPR) paradox, particles are prepared in an entangled state where their properties are correlated. According to quantum mechanics, these particles can become entangled in such a way that measuring one particle instantaneously affects the state of the other, regardless of the distance between them.
Bell's theorem and subsequent experiments inspired by it aim to test the predictions of quantum mechanics against certain assumptions made by Einstein, Podolsky, and Rosen. One of these assumptions is called "local realism" or "local hidden variables." It posits that the properties of particles have definite values even before they are measured, and any correlations between entangled particles are simply due to these pre-existing properties.
Bell's inequality is a mathematical expression that sets certain limits on the correlations that can be observed between particles if local realism is assumed. If the measurements violate Bell's inequality, it suggests that the correlations between entangled particles cannot be explained by local hidden variables, and the predictions of quantum mechanics are more accurate.
Now, coming back to your question, when testing Bell's inequality, the choice of measurement directions or axes for the particles' spins is made independently at each measurement site. This choice is typically made randomly and can be different for each measurement.
The important point is that the choice of measurement directions does not determine the spin orientations of the particles. Rather, it is a way to test the correlations between the particles' spins along different axes and see if they violate the limits set by Bell's inequality.
In this context, the test itself is not in alignment with Einstein's assumption of definite states, as it explores the correlations between particles' properties rather than assuming pre-existing definite values for those properties.
By performing experiments that test Bell's inequality, researchers have found that the observed correlations between entangled particles do not obey the limits set by local realism. These experimental results support the predictions of quantum mechanics and suggest that the nature of reality at the quantum level is inherently non-local and that certain properties of particles are not predetermined but instead arise probabilistically upon measurement.