My understanding of quantum entanglement

[Spathi]

In one book on quantum mechanics, I found a very simple analogy to CHSH inequalities, which explains their essence without delving into the details of the physical experiment. I liked this analogy, but it seems the authors got it a little confused, so I ask for clarification if it is correct in general.

Here is the description of the mentioned analogy from the book:

Each of the two remote observers - Alice and Bob - uses a device that has two buttons, labeled M and N, and a screen that can show either +1 or -1. During the experiment, Alice and Bob are unable to communicate with each other.

The source located roughly halfway between Alice and Bob sends them a couple of particles of some kind. Alice and Bob receive these particles and each insert them into their device. Then they select a random button on the device (M or N) and press it at the same time. Each device displays a value of +1 or -1, possibly related to the state of the generated particle. The entire operation described is called an event.

Both observers keep a record of the buttons they pressed and the numbers displayed. After receiving data on a large array of numbers, both parties meet and perform a correlation analysis of their records. Specifically, they estimate the value

Here, Ma, Mb, Na, Nb are the numbers that Alice and Bob receive after pressing the corresponding buttons. Each event only contributes to one of the values MaMb, MaNb, NaMb, NaNb. The book says that if | S | is greater than 2, then Bell's inequalities are violated.

This is how, if I understand it correctly, a typical experiment looks like:

I wrote a program that counts S for different algorithms of generating measurements of Alice and Bob. The following conclusions were obtained. Suppose Alice and Bob's buttons are completely random; Alice's measurement is also completely random, and Bob's measurement depends on Alice's measurement, but does not depend on the button that Alice or Bob pressed. Then S can take values from -2 to 2 (after averaging a large number of events).

Now suppose Alice's measurement is random, and Bob's measurement is defined as follows: if Alice pressed M and Bob pressed N, Bob's measurement is opposite to Alice's, otherwise Bob's measurement is the same as Alice's. Then S equals 4 - this is a violation of Bell's inequalities.

This leads to very interesting conclusions, but only under the condition that this whole analogy is correct. For the latter algorithm (for which S = 4) Bob's measurement indirectly depends on Alice's button, but does not correlate with it; therefore, Alice cannot convey information to Bob by pressing the button for a reason. This is consistent with what I have read in various sources about quantum entanglement - it does not allow information to be transmitted, but at the same time it cannot be called a complete absence of any interaction. Einstein called it “spooky action at distance”, and this characteristic is understandable, since particles located at different times can be entangled, so “spooky action through time” is an equivalent formulation.

I suppose, this analogue corresponds to this experiment:

https://en.wikipedia.org/wiki/CHSH_inequality

As far as I can see, when the polarizer a is set with 0 degrees angle, this corresponds to pressing M by Alice, and 45 degrees correspond to pressing N by Alice; when the polarizer b is set with 22.5 degrees angle, this corresponds to pressing M by Bod, and 67.5 degrees correspond to pressing N by Bob.

Please comment on whether my post is correct.