The phase space modification associated with a nonvanishing effective mass for the primary gluons, Mg=0.66±0.08 GeV for the Jψ and Mg=1.17±0.08 GeV for the Υ, is shown to be crucial for a consistent description of the photon spectrum from their radiative decays and for the determination of αs from the recent, precise quarkonia decay branching ratios. In this approach, the role of the relativistic corrections is marginal and, after applying the Mg-dependent corrections, a good agreement is obtained with the relative perturbative running, αs(mc)∼0.30±0.02 and αs(mb)∼0.21±0.01, and with the extrapolation from deep inelastic scattering. On the other hand, for Mg=0, the analysis of all experimental c¯c and b¯b quarkonia branching ratios is consistent with the same effective value of the strong interaction coupling constant αeffs∼0.185±0.010. By assuming a "genuine," i.e., process-independent, gluon mass (∼ 1.2 GeV or higher) to be dynamically generated one predicts a strong suppression of the gluon splitting process at the Jψ and the hadronic final states should be mainly produced via gluon fusion into light q¯q pairs thus effectively reducing the fitted value of Mg from the photon spectrum in Jψ→γ+X. The gluon fusion mechanism allows us to explain the structure of the hadronic final states observed in Jψ decays and their close similarity to the continuum e+e−→ hadrons annihilation at comparable center of mass energies.