A concrete example will help: Take as our real Lie algebra
$\mathfrak{g} = \mathfrak{su}_{1,2} := \lbrace
\begin{pmatrix}
a+bi & c+di & ei\\
f+gi & -2bi & -c+di\\
hi & -f+gi & -a+bi
\end{pmatrix} : a, ..., h \in \mathbb{R} \rbrace$.
Its complexification $\mathfrak g_{\mathbb C}$ is isomorphic to $\mathfrak{sl}_3(\mathbb C)$ i.e. those $3\times 3$ complex matrices with trace $0$. In Araki's terminology, we can choose as $\mathfrak h_{\mathbb C}$ the diagonal matrices in $\mathfrak{sl}_3(\mathbb C)$. If
$\alpha_1 : diag(x_1,x_2,x_3) \mapsto x_1-x_2$ and $\alpha_2 : diag(x_1,x_2,x_3) \mapsto x_2-x_3$
then the root system for this $\mathfrak h_{\mathbb C}$ contains exactly the roots in $\mathfrak r = \{\pm \alpha_1, \pm \alpha_2, \pm (\alpha_1 +\alpha_2)\}$.
Now, Araki's $\mathfrak h_0$ (defined in 2.3) is just the traceless diagonal matrices with all real entries. Note that this is not a subset of the $\mathfrak g$ we started with. In fact,
$$\mathfrak h_0 \cap \mathfrak g = \{ \begin{pmatrix}
a & 0 & 0\\
0 & 0 & 0\\
0 & 0 & -a
\end{pmatrix} : a \in \mathbb{R} \rbrace = \mathfrak h_0^-$$
while
$$\mathfrak h_0^+ = \{ \begin{pmatrix}
c & 0 & 0\\
0 & -2c & 0\\
0 & 0 & c
\end{pmatrix} : c \in \mathbb{R} \rbrace .$$
(You can check for yourself that complex conjugation on $\mathfrak g_{\mathbb C}$, under the iso $\mathfrak g_C \simeq \mathfrak{sl}_3(\mathbb C)$, operates via
$$\sigma (\begin{pmatrix}
x_1 & x_2 & x_3\\
x_4 & x_5 & x_6\\
x_7 & x_8 & x_9
\end{pmatrix}) = \begin{pmatrix}
-\bar x_9 & -\bar x_6 & -\bar x_3\\
-\bar x_8 & -\bar x_5 & -\bar x_8\\
-\bar x_7 & -\bar x_6 & -\bar x_1
\end{pmatrix}.)$$
Now look at page 6 of the paper. In the proof of prop 2.1, we "identify $(\mathfrak h_0^-)^\ast$ with a subspace of $\mathfrak h_0^\ast$ which is the annihilator of $\mathfrak h_0^+$."
Now $\mathfrak h_0^\ast$ is just the real span of the roots, and in our example, the annihilator of $\mathfrak h_0^+$ is the one-dimensional subspace in there spanned by $\alpha_1 +\alpha_2$. That is, with the notation of the paper,
$\psi \in \mathfrak r_\psi$ if and only if $\psi$ annihilates $\mathfrak h_0^+$.
In our case, this is true for $\psi = \pm (\alpha_1 + \alpha_2)$, but not for any other root. In fact, using Araki's notation on our example, we have
$$\mathfrak r_0 = \{\alpha \in \mathfrak r: \alpha_{\vert \mathfrak h_0^-} = 0\} = \emptyset$$
and if we call $\beta := \alpha_1 + \alpha_2$ then $\mathfrak r^-$, the restrictions of all roots to $\mathfrak h_0^-$, consists of
$$\mathfrak r^- = \{\pm \beta, \pm \frac12 \beta \},$$
the latter being the respective restrictions of both $\pm \alpha_1$ and $\pm \alpha_2$.
In fact, $\sigma(\alpha_1) = \alpha_2$, and you can see now that as said above, $\beta \in r_\beta$ and we are in case a of bottom of page 6. But for $\psi=\frac12 \beta$ we have
$$\mathfrak r_{\frac12 \beta} = \{\alpha_1, \alpha_2\}$$
and here we are in case c at the top of page 7. (Case b does not occur in this example.)
The propositions on page 9 should make sense now too.
I treated very similar things in my thesis, in particular this is basically example 3.2.9 in there. Araki's $r^-$ is equivalent to the standard definition of "$k$-rational root systems" (cf. https://math.stackexchange.com/a/3349137/96384 and links from there; in the example above, we see $\mathfrak r^-$ is of type $BC_1$) although I, following others (Tits and Satake in particular), would define it slightly differently: I do not like that identification of a quotient with a subspace too much. Quotients of root systems are not yet en vogue, but under the name "foldings" they are gaining popularity, or so I like to believe.
