FALSE: agreement on an arbitrary generating family determines a measure

Define $\mu$ by masses $1\mathrm{/}2$ at $NW,SE$ and $0$ at the other points, and define $\nu$ by masses $1\mathrm{/}2$ at $NE,SW$ and $0$ at the other points. Finite summation over subsets makes both probability measures on $\mathcal{P}(X)$.

Intersections of a row and a column give every singleton, so ${\sigma }_X(\mathcal{G})=\mathcal{P}(X)$; but those singleton intersections are not themselves in $\mathcal{G}$, so $\mathcal{G}$ is not a pi-system.

Each row and each column contains one point of mass $1\mathrm{/}2$ for each measure, so $\mu$ and $\nu$ agree on every member of $\mathcal{G}$.

The measures differ, for example $\mu (\{NW\})=1\mathrm{/}2$ while $\nu (\{NW\})=0$. Together with steps 2.1 and 1.2 this refutes the claim and identifies the missing intersection-closure hypothesis.