We study the effects of the nonlinear piezoelectricity and the In distribution on the exciton energy, the electronhole electric dipole moment, and the fine-structure splitting in stress-tunable In(Ga)As/GaAs quantum dots integrated onto a piezoelectric actuator. In particular, we investigate in detail the contributions of various elements of the expansion of the electrical polarization in terms of externally induced elastic strain on the latter two important quantum dot properties. Based on the comparison of the effects of first- and second-order piezoelectricity we provide a simple relation to estimate the influence of applied anisotropic stress on the quantum dot dipole moment for quantum dots significantly lattice mismatched to the host crystal.