It is a classical result of the theory of the moduli of curves, that the stack $\bar{\mathcal{M}}_{g,n}$ of nodal curves with log-structure coming from the boundary divisor, and ${\mathcal{M}}_{g,n}^{logbas}$ of log-smooth curve with minimal log structure are equivalent. Sometimes this is summarized by the slogan that $\bar{\mathcal{M}}_{g,n}$ and ${\mathcal{M}}_{g,n}^{logbas}$ are "the same". I am trying to understand this statement in detail.
If I have a log-scheme $(S,M_S)$ (for this example it is perfectly fine to assume $S=Spec(k)$) and a nodal curve $C \to S$, it is possible to give $C$ different minimal log-structures by assigning to each node different "lengths", so elements from $M_S$. As on the level of schemes there is no difference, I have understood this in the way, that the underlying map of schemes $\underline{f}:\underline{S} \to \bar{\mathcal{M}}_{g,n}$ is the same for all different choices of length, and the difference is constituted by the map of monoids $f^*M_{\bar{\mathcal{M}}_{g,n}} \to M_S$.
If I fix a zero dimensional stratum in $\bar{\mathcal{M}}_{g,n}$, for example one of the strata in $\bar{\mathcal{M}}_{0,4}$, the moduli space of curves of the type as this stratum is just $Spec(k)$. But for $n$, the number of nodes in this stratum and $P=\mathbb{N}\pi_1 \oplus \dots \oplus \mathbb{N} \pi_n$, the log scheme $Spec(P \to k[P])$ is a moduli space for log-curves of the strata as \begin{align*} Hom_{LSch/k}(S, Spec(P \to k[P])) = Hom_{Mon}(P, \Gamma(S,M_S)) \end{align*} which is exactly assigning to each node $\pi_i$ a "length". But on the level of schemes, $Spec(P \to k[P])=\mathbb{A}^n$, which is clearly not $Spec(k)$, which contradicts my understanding that the underlying schemes are the same.
I would be very happy if someone could point out the mistake in this argument to me.
