We present new luminosity-metallicity (LZ ; M_V_-[Fe/H] and M_G_-[Fe/H]) and relations and, for the first time, empirical, Gaia three-band (G, G_BP_, G_RP_) period-Wesenheit-metallicity (PWZ) relations of RR Lyrae stars (RRLs) derived using a hierarchical Bayesian approach and new accurate parallaxes published for these variables in the Gaia Early Data Release 3 (EDR3). In a previous study, we obtained Bayesian hierarchically derived LZ relations from a sample of about four hundred Milky Way field RRLs with G-band light curves and trigonometric parallaxes published in the Gaia Data Release 2 (DR2), using V mean magnitudes, metallicities, absorptions, and pulsation periods available in the literature. We now extend that study in two directions. First, we update our previous results using trigonometric parallaxes from Gaia EDR3 and incorporate the Bayesian analysis of a first empirical PWZ relation derived using those field RRLs with G, G_BP_ and G_RP_ time-series photometry available in Gaia DR2. Secondly, we use Bayesian inference to derive LZ relations and empirical PW Gaia three-band relations from 385 RRLs belonging to 15 Milky Way globular clusters (GC) with literature-compiled spectroscopic metallicities ranging from -0.36 to -2.39 dex and prior distances extending from 2.2 to 41.2 kpc. From the samples of RRLs analysed in this paper we infer a mean Gaia EDR3 zero-point offset of -0.028 mas with median values ranging from -0.033 (LZ and PWZ models for field stars) to -0.024 mas (LZ model in the V band for GC RRLs).