I am trying to understand the intuitive idea of a minimally sufficient statistic. It is my understanding that a statistic $T$ is minimally sufficient for $\theta$ for a family of populations $X\sim P_\theta$ if any other sufficient statistic $S$ of $\theta$ is of the form $T=h(S)$ where $h$ is a borel measurable function. This makes sense in a data reduction sense as this will produce the smallest $\sigma$-algebra generated by a sufficient statistic that is a subset of the $\sigma$-algebra generated by $X$. Where I get confused is in the sense of a partition by a minimally sufficient statistic. It is my understanding that any Statistic creates a partition of the support of $X$, where the minimally sufficient partition produces classes where the probability within each class is independent of $\theta$.
What is so special about the minimally sufficient partition? I know that if we use the relation $X R Y$ iff $P_\theta(X)=G(X,Y)P_\theta(Y)$ where $G(X,Y)$ is independent of $\theta$, equivalence classes are created, but why is an equivalent criterion for minimally sufficiency be that the ratio $G(X,Y)=\frac{P_\theta(X)}{P_\theta(Y)}$ does not depend on $\theta$ $\leftrightarrow T(X)=T(Y).$
How is minimally sufficient implied by this equivalent criterion? I understand the proof but I am looking more for an intuitive understanding.