This work investigates why JahnTeller (JT) transition-metal ions like Cu2+ (d9) or Mn3+ (d4) with octahedral coordination exhibit low-symmetry distortions of the CuF64- (or MnF63-) octahedron, when Cu2+ replaces Zn2+ either in the cubic perovskite KZnF3 or in the layer perovskite K2ZnF4. The aim is to establish correlations between the splitting ?e of the parent Oh eg(x(2) y(2), 3z(2) r(2)) d orbitals and the low-symmetry local distortion ?, allowing us to unravel the local structure of Cu2+ introduced as an impurity into such perovskites. However, the question of whether tetragonal a(1g) b(1g) splitting Delta(e) of Cu2+ is proportional to Delta or ?e contains additional contributions from the rest of the lattice (crystal anisotropy) is controversial. Here we show that ?e depends linearly on Delta. High-pressure experiments and compound series involving JT ions give support for the proposed scenario and provide correlations relating Delta and ?e in Cu2+ and Mn3+ systems, from which the electronvibration coupling for the E Delta(e) JT effect is obtained experimentally.