We systematically study exclusiv2 diffractive (photo)production of vector mesons (J/ψ, ψ(2s), ϕ, and ρ) off protons in high-energy collisions and investigate whether the production is a sensitive probe of gluon saturation. We confront saturation-based results for diffractive ψ(2s) and ρ production at HERA and J/ψ photoproduction with all available data, including recent work from HERA, ALICE, and LHCb, finding good agreement. In particular, we show that the t distribution of differential cross sections of photoproduction of vector mesons offers a unique opportunity to discriminate among saturation and nonsaturation models. This is due to the emergence of a pronounced dip (or multiple dips) in the t distribution of diffractive photoproduction of vector mesons at relatively large, but potentially accessible |t| that can be traced back to the unitarity features of color dipole amplitude in the saturation regime. We show that in saturation models the dips in t distribution recede towards lower |t| with decreasing mass of the vector meson, increasing energy or decreasing Bjorken-x, and decreasing virtuality Q. We provide various predictions for exclusive (photo)production of different vector mesons including the ratio of ψ(2s)/J/ψ at HERA, the LHC, and future colliders.