In Silverman's "Advanced topics in elliptic curves", To prove Tate's algorithm, lemma 9.5 in chapter 4.9 states:
Let $R$ be a DVR with fraction field $K$, $E/K$ an elliptic curve with Weierstrass equation (...), $W \subset P^2_R$ the $R$-scheme defined by this equation.
a) If $v(\Delta)=1$ then $W$ is regular and $W$ is the minimal proper regular model of the elliptic curve.
However Silverman does not prove the minimality part. Why is this true? That is, why cant $W \to \operatorname{Spec} R$ be factored as $$W \to W' \to \operatorname{Spec} R$$ with the generic fiber of $W'$ isomorphic to $E/K$?
Doesn't $v(\Delta)=1$ imply (among other things) that the special fiber has a single component (a nodal curve that is birational to $\mathbb P^1$). That means that every fibral component moves, since there is only one component, so it has self-intersection $0$, so it cannot be blown down.