We study the quantum-to-classical transition in a chaotic system surrounded by a diffusive environment. First, we analyze the emergence of classicality when it is monitored by the Renyi entropy, a measure of the entanglement of a system with its environment. We show that the Renyi entropy has a transition from quantum to classical behavior that scales with heff2D, where heff is the effective Planck constant and D is the strength of the noise. However, it was recently shown that a different scaling law controls the quantum-to-classical transition when it is measured comparing the corresponding phase-space distributions. Then, we discuss the meaning of both scalings in the precise definition of a frontier between the classical and quantum behaviors. Finally, we show that there are quantum coherences that the Renyi entropy is unable to detect, which questions its use in studies of decoherence.