Definition of "properly partial" versus "total" value types
$
\newcommand{\llbracket}{[\![}
\newcommand{\rrbracket}{]\!]}
$First, a proper subset is a subset that is not equal to the whole set. In other words, $A$ is a proper subset of $B$ iff $A \subseteq B$ and $A \ne B$.

If you formulate a denotational semantics for a language, then each type $T$ has an associated set $\llbracket T \rrbracket$ which is called the denotation of $T$. Stepanov and McJones calls this set the abstract species corresponding to the type $T$ (in this particular semantics). For example, it is natural to associate the set of integers $\mathbb{Z}$ to the type $\mathtt{int}$, with value semantics $\llbracket \mathtt{0} \rrbracket = 0$, $\llbracket \mathtt{1} \rrbracket = 1$, $\llbracket \mathtt{-1} \rrbracket = -1$, etc. and the set of truth values $\{\mathsf{T},\mathsf{F}\}$ to the type $\mathtt{bool}$ with value semantics $\llbracket \mathtt{false} \rrbracket = \mathsf{F}$ and  $\llbracket \mathtt{true} \rrbracket = \mathsf{T}$.

A type is total under a denotational semantics iff the set of denotations of values of that type is equal to the denotation of the type. A type is properly partial iff is it not total, that is, iff the set of denotations of values is a proper subset of the denotation of the type. (The denotation of a value must be a member of the denotation of the type, otherwise the semantics is not valid, thus the denotations of values always form subset of the denotation of the type.)

For example, the set of denotations of values of the type $\mathtt{bool}$ under the semantics above is $\{\llbracket \mathtt{false} \rrbracket, \llbracket \mathtt{true} \rrbracket\} = \{\mathsf{F}, \mathsf{T}\} = \llbracket \mathtt{bool} \rrbracket$ so the type $\mathtt{bool}$ is total under this semantics. On the other hand, the set of denotations of values of type $\mathtt{int}$ consists only of values in the range $[-2^{N-1}, 2^{N-1}-1]$, assuming a typical implementation of the type $\mathtt{int}$ with $N$ bits, therefore $\mathtt{int}$ is properly partial under this semantics.

Note that the totality of a type depends on the chosen semantics. For example, under a semantics where the denotation of $\mathtt{int}$ is the integer range $[-2^{N-1}, 2^{N-1}-1]$, the type $\mathtt{int}$ is total.

”Abstract species“ and “properly partial” are not widespread terminology. As far as I know, they were coined by these authors.