I would like to understand how from $$d\omega_2(x)=(\frac{\partial g_{2,3}}{\partial x_1}-\frac{\partial g_{1,3}}{\partial x_2}+\frac{\partial g_{1,2}}{\partial x_3})dx_{1,2,3}$$
follows $$\operatorname{div}F=\frac{\partial F_1}{\partial x_1}+\frac{\partial F_2}{\partial x_2}+\frac{\partial F_3}{\partial x_3} $$ as claimed in the book by Fonda in the snippet below:
The divergence already has a classical definition (which you typed in your second equation). He's just arriving at the formula through the machinery of differential forms. To the vector field $F$ we associate the $2$-form $\omega = F_1dx_{23} + F_2dx_{31}+F_3dx_{12}$. Then $d\omega = (\text{div}\,F)dx_{123}$. He says this in the text if you read the lower half of the page. So in his earlier notation, he's taking $F_1=g_{23}$, $F_2=-g_{13}$, and $F_3=g_{12}$.
(You might find the latter half of my YouTube lecture helpful. Perhaps others of my lectures will be useful, as well.)