I understood that properness of $\overline{\mathcal{M}_{g,n}}$ tells us that for a fixed field $K$, a curve $C$ over $K$ and a DVR $V$ with fraction field $K$, there is a unique curve over $V$ with generic fibre $C$.
However, I have the feeling that there is some choice in smoothening a node. For example for $n \neq 0$, the generic fibre of (the completion of) $Spec(V[x,y]/(xy-\pi^n))$, where $\pi$ is a uniformizing parameter, will always be $\mathbb{P}^1_K$, but these curves are non-isomorphic as curves over $V$, as the rings $V[x,y]/(xy-\pi^n)$ are non-isomorphic for different $n$, which would mean that $\mathbb{P}^1_K$ has different lifts to $V$, which would contradict properness.
Could someone point out to me where the mistake in my argumentation is?