ABSTRACT The relationships between stellar mass, gas-phase metallicity and star-formation rate (i.e. the mass–metallicity, MZR, and the fundamental metallicity relation, FMR) in the local Universe are revisited by fully anchoring the metallicity determination for SDSS galaxies on the Te abundance scale defined exploiting the strong-line metallicity calibrations presented by Curti et al. Self-consistent metallicity measurements allow a more unbiased assessment of the scaling relations involving M, Z and SFR, which provide powerful constraints for the chemical evolution models. We parametrize the MZR with a new functional form that allows us to better characterize the turnover mass. The slope and saturation metallicity are in good agreement with previous determinations of the MZR based on the Te method, while showing significantly lower normalization compared to those based on photoionization models. The Z–SFR dependence at fixed stellar mass is also investigated, being particularly evident for highly star-forming galaxies, where the scatter in metallicity is reduced up to a factor of ${∼}30{{\ \rm per\ cent}}$. A new parametrization of the FMR is given by explicitly introducing the SFR dependence of the turnover mass into the MZR. The residual scatter in metallicity for the global galaxy population around the new FMR is 0.054 dex. The new FMR presented in this work represents a useful local benchmark to compare theoretical predictions and observational studies (of both local and high-redshift galaxies) whose metallicity measurements are tied to the abundance scale defined by the Te method, hence allowing proper assessment of its evolution with cosmic time.