The relationships between stellar mass, gas-phase metallicity and star formation rate (i.e. the Mass-Metallicity, MZR, and the Fundamental Metallcity Relation, FMR) in the local Universe are revisited by fully anchoring the metallicity determination for SDSS galaxies on the Te abundance scale de ned exploiting the strong-line metallicity calibrations presented in Curti et al. (2017). 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 paramet-rise the MZR with a new functional form which allows us to better characterise the turnover mass. The slope and saturation metallicity are in good agreement with pre- vious determinations of the MZR based on the Te method, while showing signi cantly lower normalisation compared to those based on photoionisation models. The Z-SFR dependence at xed 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%. A new parametrisation of the FMR is given by explicitly introducing the SFR-dependence of the turnover mass into the MZR. The residual scatter in metal-licity 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 theor-etical predictions and observational studies (of both local and high-redshift galaxies) whose metallicity measurements are tied to the abundance scale de ned by the Te method, hence allowing to properly assess its evolution with cosmic time.