= Solution
The <density of states> $D(E)$ counts available one-electron states: $D(E)\,dE$ is the number with energies in $[E,E+dE]$. It describes the spectrum, irrespective of whether those states contain electrons.
The <density of occupied states> additionally weights each state by the <Fermi-Dirac distribution>:
$$
\boxed{Z(E)=D(E)f(E)}.
$$
Thus $Z(E)\,dE$ is the mean number of electrons in that energy interval, and $N_e=\int Z(E)\,dE$. State and electron densities must use the same normalization, including whether they are extensive or per volume and whether $D$ already includes <spin degeneracy>. The symbol $Z(E)$ here denotes occupied density, not a canonical <partition function>.
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