Neutron stars - only neutrons?

The picture from Wikipedia is (qualitatively) correct and so is your intuition.

Neutrons are unstable and decay unless the decay is blocked by the presence of a degenerate electron gas with a Fermi energy that is as large as the maximum possible energy of the electron produced in a beta decay.

If all fermion species are degenerate, and they are at neutron star densities, then the Fermi energies of neutrons, protons and electrons come into an equilibrium where $$E_{f,n} = E_{f,p} + E_{f,e}$$ This, combined with charge conservation, leads to the calculation that there are about ten to a hundred times as many neutrons than protons/electrons in what started as a pure neutron gas; the exact ratio being density dependent (smaller at higher densities).

Aside from this, the equilibrium state of "cold" neutron star matter varies with density, as shown in the Wikipedia diagram. The n,p,e fluid probably makes up the greater part of the mass of a neutron star, but by no means can it ever be said that a neutron star is made up only of neutrons, and in fact free neutrons only appear at densities above a few $10^{14}$ kg/m$^3$, some way in to the neutron star.

NB This is where the diagram is quantitatively incorrect. $\rho_0$ is supposed to be the nuclear saturation density of about $2.3\times 10^{17}$ kg/m$^3$, so free neutrons appear at just over $10^{-3}\rho_0$. In addition, nuclei lose their identity and become an n,p,e gas at about $0.2\rho_0$.