We consider the flow of a generalized Newtonian fluid through a thin porous medium of height hε perforated with ε-periodically distributed solid cylinders of very small diameter εδε, where the small parameters ε, δε and hε are devoted to tend to zero. We assume that the fluid is described by the 3D incompressible Stokes system with a non-linear power law viscosity of flow index 1 < r < 2 (shear thinning). The particular case hε = σε, where σε := ε/δε^{(2-r)/r} → 0, was recently published in (Anguiano and Suárez-Grau, Mediterr. J. Math. (2021) 18:175). In this paper, we generalize previous study for any hε and we provide a more complete description on the asymptotic behavior of non-Newtonian fluids in a thin porous medium composed by cylinders of small diameter. We prove that depending on the value of λ := limε→0 σε/hε ∈ [0,+∞], there exist three types of lower-dimensional asymptotic models: a non-linear Darcy law in the case λ = 0, a non-linear Brinkman-type law in the case λ ∈ (0, +∞), and a non-linear Reynolds law in the case λ = +∞.