Your first equation \nabla p = J \times B is only a general balance equation, the name 'GS equ.' actually only applies to this one:
\Delta^{*}\psi = -\mu_{0}R^{2}\frac{dp}{d\psi}-\frac{1}{2}\frac{dF^2}{d\psi}
It is derived from the balance equation, but assumes axisymmetry.As you mentioned...