From analysis of the in-plane resistivity ρab(T) of La2−xSrxCuO4, we show that normal state transport in overdoped cuprates can be delineated into two regimes in which the electrical resistivity varies approximately linearly with temperature. In the low-temperature limit, the T-linear resistivity extends over a very wide doping range, in marked contrast to expectations from conventional quantum critical scenarios. The coefficient of this T-linear resistivity scales with the superconducting transition temperature Tc, implying that the interaction causing this anomalous scattering is also associated with the superconducting pairing mechanism. At high temperatures, the coefficient of the T-linear resistivity is essentially doping independent beyond a critical doping pcrit=0.19 at which the ratio of the two coefficients is maximal. Taking our cue from earlier thermodynamic and photoemission measurements, we conclude that the opening of the normal-state pseudogap at pcrit is driven by the loss of coherence of anti-nodal quasi-particles at low temperatures.