  We report on an extensive study of the elastic scattering time $\tauS$ of
matter-waves in optical disordered potentials. Using direct experimental
measurements, numerical simulations and comparison with first-order Born
approximation based on the knowledge of the disorder properties, we explore the
behavior of $\tauS$ over more than three orders of magnitude, spanning from the
weak to the strong scattering regime. We study in detail the location of the
crossover and, as a main result, we reveal the strong influence of the disorder
statistics, especially on the relevance of the widely used Ioffe-Regel-like
criterion $k\lS\sim 1$. While it is found to be relevant for
Gaussian-distributed disordered potentials, we observe significant deviations
for laser speckle disorders that are commonly used with ultracold atoms. Our
results are crucial for connecting experimental investigation of complex
transport phenomena, such as Anderson localization, to microscopic theories.
