We improve and extend our study of the complex in-medium heavy-quark potential and its Debye mass m(D) in a gluonic medium with a finer scan around the deconfinement transition and newly generated ensembles closer to the thermodynamic limit. On the lattices with larger physical volume, Re [V] shows signs of screening, i.e. a finite m(D), only in the deconfined phase, reminiscent of a genuine phase transition. Consistently Im[V] exhibits nonzero values also only above T-C. We compare the behavior of Re [V] with the color singlet free energies that have been used historically to extract the Debye mass. An effective coupling constant is computed to assess the residual influence of the confining part of the potential at T > 0. Our previous finding of a gradual screening of Re [V] around T-C on finer lattices is critically reassessed and interpreted to originate from finite volume artifacts. We discuss that deficiency of the beta = 7, xi(b) = 3.5 parameter set at N-s = 32, which has been in deployed in the literature before.