It states that no two electrons can have the same four quantum numbers.
In the same atom?
In the same solid object?
In the same universe?
In your example, a single atom. The sentence you note from the article mentions this:
(Emphasis added.) The more general principle applies to all identical fermions, not just those in a given atomic bound state for which these four quantum numbers are relevant. Note that an electron in a neighboring atom can have the same quantum numbers because it has a distinct spatial wavefunction. That is, those four quantum numbers don’t specify the state fully. One must also say where the electron is, and “in this atom” is different from “in that atom”.
Aha…I missed that.
If you take an ensemble wavefunction \psi and swap two identical particles in it, then you should get the same physics. That means that that interchange S acts on \psi by multiplication by some constant c; and since S^2 is trivial, c^2 = 1; that is, c = +1 or -1. It turns out that c = +1 for bosons (particles with integer spin) and c = -1 for fermions (particles with half-integer spin). If \psi has identical fermions in the same state, then swapping them maps \psi -> -\psi; but if they’re identical and in the same state, then swapping them obviously leaves \psi invariant. Oops.
Electrons in a given atom are characterized by their quantum numbers. (Three of these just parameterize reasonable solutions to the Schrodinger equation for an atom; the spin number is a bit trickier). Electrons have spin 1/2, so no two electrons in the same atom can have the same set of quanutm numbers. Electrons in different atoms can— and certainly do, because there are lots of atoms and few quantum numbers with low energy levels. They’re not lumped together in the same ensemble wavefunction, so the argument above doesn’t apply.