Let me explain how both kinds of foldings of Dynkin diagrams (i.e., $A_{2n-1} \to B_n$ and $A_{2n-1} \to C_n$) arise in the context of Lie algebras and characters of their representations.
First of all, what I will call the "standard combinatorial procedure" for folding a root system $\Phi$ according to a Dynkin diagram automorphism $\sigma$, as described by Stembridge here, will produce the Type $B_n$ diagram from a Type $A_{2n-1}$ diagram (and will produce a Type $C_n$ diagram from a Type $D_{n+1}$ diagram). The standard procedure is to produce the root system whose simple roots $\beta_{I}$ correspond to orbits $I \subseteq \Delta$ of the simple roots of the original diagram under the automorphism $\sigma$: we just take the sum of the roots in each orbit $\beta_{I} := \sum_{\alpha\in I} \alpha$. Let me call this folded diagram root system $\Phi^{\sigma}$.
However, if $\mathfrak{g}$ is the Lie algebra of $\Phi$, then the automorphism $\sigma$ acts on $\mathfrak{g}$ in an obvious way, and the fixed-point Lie subalgebra $\mathfrak{g}^{\sigma}$ has as its root system the dual of $\Phi^{\sigma}$. In this way we get the inclusions $\mathfrak{sp}_{2n}\subseteq \mathfrak{sl}_{2n}$, and $\mathfrak{so}_{2n} \subseteq \mathfrak{so}_{2n+1}$ (i.e. $A_{2n-1} \to C_n$ and $D_{n+1} \to B_n$).
But there is way to make the "standard" folded root system $\Phi^{\sigma}$ appear in the context of Lie algebras as well, namely, by considering so-called "twining characters." Let me call the Lie algebra associated to $\Phi^{\sigma}$ the "orbit Lie algebra" of $(\Phi,\sigma)$.
The set-up in which the orbit Lie algebra arises is this: we can "twist" any representation $V$ of $\mathfrak{g}$ by the automorphism $\sigma$ to get a new, twisted representation $V^{\sigma}$; if $V=V^{\lambda}$ is the highest-weight representation with highest weight $\lambda$, then $V^{\sigma} = V^{\sigma(\lambda)}$, where $\sigma$ acts on the weight lattice of $\Phi$ in the obvious way. Suppose that we choose a $\sigma$-fixed weight $\lambda$. Then we can view $\sigma$ as a map $\sigma\colon V^{\lambda}\to V^{\lambda}$ (I think this is technically defined up to scalar). The twining character of $V^{\lambda}$ is defined to be $\mathrm{ch}^{\sigma}(V^{\lambda})(h) = \mathrm{tr}(\sigma\cdot e^{h})$ for $h \in \mathfrak{h}$, just like the usual character would be $\mathrm{ch}(V^{\lambda})(h) = \mathrm{tr}(e^{h})$. The twining character formula, which is originally due to Jantzen (see the discussion at the beginning of https://arxiv.org/abs/1404.4098) but has been rediscovered by many people (e.g., https://arxiv.org/abs/hep-th/9612060, https://arxiv.org/abs/q-alg/9605046), asserts that the twining character $\mathrm{ch}^{\sigma}(V^{\lambda})$ is equal to the usual character $\mathrm{ch}(U^{\lambda})$ where $U^{\lambda}$ is the highest-weight representation of the orbit Lie algebra with highest weight $\lambda$ (note that since $\lambda$ is fixed by $\sigma$, it is naturally a weight of the folded root system $\Phi^{\sigma}$). So the upshot is that a Type $A_{2n-1}$ twining character is a Type $B_{n}$ ordinary character.
The twining characters have some interesting applications to combinatorics when considering "symmetric" versions of combinatorial objects associated to Lie algebras, which is how I became aware of them. I quote from the 2nd column of the 4th page of this paper of Kuperberg (https://arxiv.org/abs/math/9411239):
As stated in the proof, $\sigma_B$ is a Dynkin diagram automorphism. The character theory of semi-direct products arising from Dynkin diagram automorphisms is described by Neil Chriss [2], who explained to the author that although this theory is known to several representation theorists, it may not have been previously published. The group $\mathbb{Z}/2 \ltimes_{\sigma_B} SL(2a)$ has two components. The character of a representation on the identity component is just the usual character of $SL(2a)$. The character on the $\sigma_B$ component, when non-zero, equals the character of an associated representation of the dual Lie group, in this case $SO(2a + 1)$, to the subgroup fixed by the outer automorphism, in this case $Sp(2a)$. The representation associated to $V_{SL(2a)}(c\lambda_a)$ is the projective representation $V_{SO(2a+1)}(c\lambda_a)$, where $\lambda_a$ is now the weight corresponding to the short root of $B_a$, the root system of $SO(2a + 1)$. In particular, the trace of $\sigma_B$ is the dimension of $V_{SO(2a+1)}(c\lambda_a)$, as given by the Weyl dimension formula, and the trace of $\sigma_BD_q$ is the q-dimension, as given by the Weyl q-dimension formula.
The disclaimer at the beginning of this quote suggests that (at least in 1994) this folding business was not well-known or written down precisely in a canonical text.
I believe the right reference is Borel-de Siebenthal.
A finite-dimensional proof is as follows. The space of conjugacy classes $G/\sim$ can be identified with $T/W = (Lie(T)/\Lambda)/W = Lie(T)/(\Lambda \rtimes W) = Lie(T)/{\widehat W}$, where $\Lambda = ker(\exp:Lie(T)\to T)$, and the final equality uses $G$ simply connected. Then this last space is the Weyl alcove $\Delta$.
There's an obvious action of $Z(G)$ on $G/\sim$, and the corresponding action on $\Delta$ is by isometries. In particular, it takes corners to corners. Some of those corners are in the orbit of the identity. So far we have an injection $Z(G) \to$ {corners of $\Delta$}, which of course then bijects to vertices of the affine Dynkin diagram. Lastly, we have to figure out the image.
Given a point $t \in T$ lying over some point $p$ in the relative interior of a $k$-face of $\Delta$, the centralizer $C_G(t)$ has central rank $k$, semisimple rank $rank(G)-k$. If $p$ is a corner, then $C_G(t)$ is semisimple, and the neighborhood of $p$ in $\Delta$ is the Weyl chamber of $C_G(t)$. So the root lattice of $C_G(t)$ is finite index in that of $G$, and this index is the coefficient of the corresponding simple root in the affine root.
The center is mapping to those vertices for which $C_G(t) = G$, i.e. this coefficient is therefore $1$.
Best Answer
(Maybe this is an answer, maybe it merely moves the lump around under the carpet.) M. Artin showed ["On rational singularities of surfaces"] that for any rational surface singularity (in particular, for du Val singularities) the fundamental cycle $Z$ (by definition, the smallest non-zero effective divisor $Y$ supported on the exceptional locus of the minimal resolution $X$ such that $Y.C\le 0$ for all exceptional curves $C$ on $X$) is defined, as a scheme, by the maximal ideal of the original singularity. Therefore $X$ factors through the first blow-up and the exceptional curves that appear there are exactly those whose strict transforms on $X$ have strictly negative intersection with $Z$.
Now think of $Z$ merely as an element of the appropriate irreducible root lattice, bearing in mind that in this context the roots $\alpha$ have square (= self-intersection) $\alpha.\alpha$ equal to $-2$. Then $Z$ is characterized by the property that $Z$ is positive linear combination of the simple roots $\alpha_i$, the intersection number $Z.\alpha_i\le 0$ for each $i$ and $Z$ is minimal with respect to these conditions. Separate consideration of each of these (I don't know how to make this step "natural") shows that $Z$ equals the highest root ("plus grande racine"). Therefore the irreducible curve(s) that appear in the first blow-up correspond to the simple root(s) $\alpha_j$ such that $Z.\alpha_j<0$. In turn, further inspection of each Dynkin diagram shows that such $\alpha_j$ correspond to the node(s) that "you would connect to the additional node on an extended Dynkin diagram", exactly as you say.