A systematic study of the graphene growth kinetics on Cu during chemical vapor deposition (CVD) by methane decomposition extends the predictive kinetic description to the medium- and low-pressure regimes. In both pressure regimes, a pre-equilibrated adsorbate phase feeds graphene island growth, resulting in a constant flake diameter increase per time of each growing island independent of its size. At reaction pressures above ≈10 mbar, the graphene growth proceeds from a pre-equilibrated C-ad phase, whereas the kinetics change toward growth fed by a pre-equilibrated CH₂-ad phase when lowering the pressure of the reactive gases. For both CVD regimes, quantitative predictions of the graphene flake size increase per time and also the transition between the different growth kinetics can be achieved for a large parameter (p(CH₄), p(H₂), T) range. The validity of the kinetic description is verified by comparison with literature data. The gained insight regarding the nature of the feeding adsorbate phase is discussed with the help of an enthalpy diagram of the CH₄ decomposition processes over copper leading to graphene. For this purpose, an appropriate energy diagram is constructed from experimental and theoretical work while referencing to the relevant conditions of a CVD synthesis experiment.