Spin up and down have no meaning in an absolute sense, so in the example you attached, flipping the spin of the whole system does not change any physics, since there is no reference magnetic field that would mean spin up/down have different energy. This is where the fixed spin comes in, it can sort of mimic a magnetic field.
Relative spins do matter, and can be used in noncollinear and collinear. So if you had the two electrodes with opposite spins, or same spin, these would be different physical situations, with different transport (parallel and antiparallel, as needed to compute TMR for instance).