In the solution all partial derivatives are evaluated at the equilibrium point
Why does the solution not talk about the fact that the determinant of the Jacobian Matrix=$f_ug_v-f_vg_u$ at the equilibrium point must be $>0$, for a turing instability to occur?
I'm afraid the solution is rather sloppy by failing to mention that. As you say, for a Turing bifurcation to occur, the steady state must be stable with respect to spatially homogeneous perturbations, which means that the determinant of the Jacobian must be positive, and its trace must be negative. In this case, the determinant of the Jacobian is $(a+b)^2$, which is always positive.