It's like the quantum number for spin-up/down was reset during the spin-left/right test. (reset from "up" to "?", please excuse my layman's language)
Yes. The notes have not explicitly reached this point, but that is what will be developed next. Testing x-spin resets z-spin to "?". We say they are incompatible observables, they cannot both possess definite values. (This will turn out to be the origin of the uncertainty principle.)
Most of the mathematics really boils down to precisely quantifying the many forms "?" can take and getting quantitative predictions from it.
In the "B" experiment this detection of spin-left/right is not made and the result of the second up/down test yields 100% "up". Even though the beam in the spin-left/right apparatus is split into separate spin-left and spin-right paths (since the equipment is exactly the same, functions just the same (minus the detector)) the quantum number ("up") from the first up/down test block was not reset by the left/right test block.
The only difference between the setup of the original and "A" experiments and the setup of the "B" experiment is that the original and "A" detect the left/right split while "B" does not.
Exactly. Essentially eliminating anything but the detection process as the source of the difference.
This is a "Double Slit" experiment for electron spin, yes?
Yes, it is the direct analogue of the Double slit experiment for spin. The unfortunate thing about the double slit is that to describe it you need advanced mathematics (Hilbert spaces, complex-valued partial differential equations). However for spin, the mathematics is far simpler, vectors and matrices.
So in a layman's analogy (not totally accurate but close enough for some understanding) is saying ...
"When the spin-left/right paths are detected the quantum number for spin-up/down is reset to "?" thus allowing the second up/down test to spread the results among the quantum probabilities" ...
Actually that's totally accurate. We only need to develop an understanding of the "?" state, i.e. are there different types of "?", how do we describe them mathematically.