You have a very thin cylinder. Ellipsoid seems to perform better if you are trying to emulate a disk.
cyl = Cylinder[{{(-0.1 + 10.128)/2, (13 + 1.3)/2, 0}, {(-0.1 + 10.128)/2, (13 + 1.3)/2, 0.001}}, 3.2]; centroid = RegionCentroid@cyl; bnds = First@Differences@Transpose@RegionBounds@cyl; ell = Ellipsoid[centroid, bnds]; Manipulate[ Graphics3D[{Specularity[White, 5], Style[ell, c]}, Lighting -> {{"Spot", Red, {centroid + {0, 0, 3}, centroid}, θ}}, ImageSize -> Small], {{θ, π/2.5}, π/40, π/2}, {c, White, ColorSlider}]

The following shows how the image changes at discrete surface colors with constant lighting.
Grid[{Graphics3D[{Style[ell, #]}, Lighting -> {{"Spot", Red, {centroid + {0, 0, 3}, centroid}, Pi/2.5}}] & /@ {White, Yellow, LightBlue, Blue}}]

The following shows how the image changes with underlying surface color under a White, Red, Green, and Blue spotlight.
mean = First@Mean@Transpose@RegionBounds@cyl; tt = TranslationTransform /@ (DeleteCases[ Tuples[{0, #}, 3], {_, _, #}] &@bnds[[1]]*2.5); ctt = Transpose[{{"White", "Red", "Blue", "Green"}, tt}]; Manipulate[ Graphics3D[{Specularity[White, 5], Flatten@({Lighting -> {{"Spot", ToExpression@#[[1]], #[[2]] /@ {centroid + {0, 0, 3}, centroid}, s}}, GeometricTransformation[ Style[{ell, Text[#[[1]], {mean, bnds[[1]]*2.3, 0}]}, c], #[[ 2]]]} & /@ ctt)}, ViewPoint -> Top], {{s, Pi/5, "Spotlight Angle"}, Pi/20, Pi/2.5}, {{c, Yellow, "Underlying Object Color"}, ColorSlider}, ControlPlacement -> Top]
