Let $X$ be some minimal model of $C$. The numerical type associated to $X$ has genus $0$ and is minimal (Definition \ref{definition-numerical-type-model} and Lemma \ref{lemma-numerical-type-minimal-model}). Hence by Lemma \ref{lemma-genus-zero} we see that $X_k$ is reduced, irreducible, has $H^0(X_k, \mathcal{O}_{X_k}) = k$, and has genus $0$. Let $Y$ be a second minimal model for $C$ which is not isomorphic to $X$. By Resolution of Surfaces, Lemma \ref{resolve-lemma-birational-regular-surfaces} there exists a diagram of $S$-morphisms $$ X = X_0 \leftarrow X_1 \leftarrow \ldots \leftarrow X_n = Y_m \to \ldots \to Y_1 \to Y_0 = Y $$ where each morphism is a blowup in a closed point. We will prove the lemma by induction on $m$. The base case is $m = 0$; it is true in this case because we assumed that $Y$ is minimal hence this would mean $n = 0$, but $X$ is not isomorphic to $Y$, so this does not happen, i.e., there is nothing to check. \medskip\noindent Before we continue, note that $n + 1 = m + 1$ is equal to the number of irreducible components of the special fibre of $X_n = Y_m$ because both $X_k$ and $Y_k$ are irreducible. Another observation we will use below is that if $X' \to X''$ is a morphism of regular proper models for $C$, then $X' \to X''$ is an isomorphism over an open set of $X''$ whose complement is a finite set of closed points of the special fibre of $X''$, see Varieties, Lemma \ref{varieties-lemma-modification-normal-iso-over-codimension-1}. In fact, any such $X' \to X''$ is a sequence of blowing ups in closed points (Resolution of Surfaces, Lemma \ref{resolve-lemma-proper-birational-regular-surfaces}) and the number of blowups is the difference in the number of irreducible components of the special fibres of $X'$ and $X''$. \medskip\noindent Let $E_i \subset Y_i$, $m \geq i \geq 1$ be the curve which is contracted by the morphism $Y_i \to Y_{i - 1}$. Let $i$ be the biggest index such that $E_i$ has multiplicity $> 1$ in the special fibre of $Y_i$. Then the further blowups $Y_m \to \ldots \to Y_{i + 1} \to Y_i$ are isomorphisms over $E_i$ since otherwise $E_j$ for some $j > i$ would have multiplicity $> 1$. Let $E \subset Y_m$ be the inverse image of $E_i$. By what we just said $E \subset Y_m$ is an exceptional curve of the first kind. Let $Y_m \to Y'$ be the contraction of $E$ (which exists by Resolution of Surfaces, Lemma \ref{resolve-lemma-contract-when-quasi-projective}). The morphism $Y_m \to X$ has to contract $E$, because $X_k$ is reduced. Hence there are morphisms $Y' \to Y$ and $Y' \to X$ (by Resolution of Surfaces, Lemma \ref{resolve-lemma-factor-through-contraction}) which are compositions of at most $n - 1 = m - 1$ contractions of exceptional curves (see discussion above). We win by induction on $m$. Upshot: we may assume that the special fibres of all of the curves $X_i$ and $Y_i$ are reduced. \medskip\noindent Since the fibres of $X_i$ and $Y_i$ are reduced, it has to be the case that the blowups $X_i \to X_{i - 1}$ and $Y_i \to Y_{i - 1}$ happen in closed points which are regular points of the special fibres. Namely, if $X''$ is a regular model for $C$ and if $x \in X''$ is a closed point of the special fibre, and $\pi \in \mathfrak m_x^2$, then the exceptional fibre $E$ of the blowup $X' \to X''$ at $x$ has multiplicity at least $2$ in the special fibre of $X'$ (local computation omitted). Hence $\mathcal{O}_{X''_k, x} = \mathcal{O}_{X'', x}/\pi$ is regular (Algebra, Lemma \ref{algebra-lemma-regular-ring-CM}) as claimed. In particular $x$ is a Cartier divisor on the unique irreducible component $Z'$ of $X''_k$ it lies on (Varieties, Lemma \ref{varieties-lemma-regular-point-on-curve}). It follows that the strict transform $Z \subset X'$ of $Z'$ maps isomorphically to $Z'$ (use Divisors, Lemmas \ref{divisors-lemma-strict-transform} and \ref{divisors-lemma-blow-up-effective-Cartier-divisor}). In other words, if an irreducible component $Z$ of $X_i$ is not contracted under the map $X_i \to X_j$ ($i > j$) then it maps isomorphically to its image. \medskip\noindent Now we are ready to prove the lemma. Let $E \subset Y_m$ be the exceptional curve of the first kind which is contracted by the morphism $Y_m \to Y_{m - 1}$. If $E$ is contracted by the morphism $Y_m = X_n \to X$, then there is a factorization $Y_{m - 1} \to X$ (Resolution of Surfaces, Lemma \ref{resolve-lemma-factor-through-contraction}) and moreover $Y_{m - 1} \to X$ is a sequence of blowups in closed points (Resolution of Surfaces, Lemma \ref{resolve-lemma-proper-birational-regular-surfaces}). In this case we lower $m$ and we win by induction. Finally, assume that $E$ is not contracted by the morphism $Y_m \to X$. Then $E \to X_k$ is surjective as $X_k$ is irreducible and by the above this means it is an isomorphism. Hence $X_k$ is isomorphic to a projective line as desired.