Abstract: The magma ocean (MO) is a crucial stage in the build-up of terrestrial planets. Its solidification and the accompanying outgassing of volatiles set the conditions for important processes occurring later or even simultaneously, such as solid-state mantle convection and atmospheric escape. To constrain the duration of a global-scale Earth MO, we have built and applied a 1D interior model coupled with either a gray H2O/CO2 atmosphere or with a pure H2O atmosphere treated with a line-by-line model described in a companion paper by Katyal et al. We study in detail the effects of several factors affecting the MO lifetime, such as the initial abundance of H2O and CO2, the convection regime, the viscosity, the mantle melting temperature, and the longwave radiation absorption from the atmosphere. In this specifically multivariable system, we assess the impact of each factor with respect to a reference setting commonly assumed in the literature. We find that the MO stage can last from a few thousand to several million years. By coupling the interior model with the line-by-line atmosphere model, we identify the conditions that determine whether the planet experiences a transient MO or it ceases to cool and maintains a continuous MO. We find a simultaneous dependence of this distinction on the mass of the outgassed H2O atmosphere and on the MO surface melting temperature. We discuss their combined impact on the MO's lifetime in addition to the known dependence on albedo, orbital distance, and stellar luminosity, and we note observational degeneracies that arise thereby for target exoplanets.