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We reveal the generic characteristics of wave-packet delocalization in two-dimensional nonlinear disordered lattices by performing extensive numerical simulations in two basic disordered models: the Klein-Gordon system and the discrete nonlinear Schrödinger equation. We find that in both models (a) the wave packet's second moment asymptotically evolves as tam with am ≈ 1/5 (1/3) for the weak (strong) chaos dynamical regime, in agreement with previous theoretical predictions [S. Flach, Chem. Phys. 375, 548 (2010)]; (b) chaos persists, but its strength decreases in time t since the finite-time maximum Lyapunov exponent Λ decays as Λ ∝ tαΛ, with αΛ ≈ -0.37 (-0.46) for the weak (strong) chaos case; and (c) the deviation vector distributions show the wandering of localized chaotic seeds in the lattice's excited part, which induces the wave packet's thermalization. We also propose a dimension-independent scaling between the wave packet's spreading and chaoticity, which allows the prediction of the obtained α Λ values. [ABSTRACT FROM AUTHOR]