What determines the nuclear radio emission in local galaxies? To address this question, we combine optical [O III] line emission, robust black hole (BH) mass estimates, and high-resolution e-MERLIN 1.5-GHz data, from the LeMMINGs survey, of a statistically complete sample of 280 nearby optically active (LINER and Seyfert) and inactive [H II and absorption line galaxies (ALGs)] galaxies. Using [O III] luminosity (L[O III]) as a proxy for the accretion power, local galaxies follow distinct sequences in the optical-radio planes of BH activity, which suggest different origins of the nuclear radio emission for the optical classes. The 1.5-GHz radio luminosity of their parsec-scale cores (Lcore) is found to scale with BH mass (Mbh) and [O III] luminosity. Below Mbh ~ 10^(6.5) Msun, stellar processes from non-jetted H II galaxies dominate with Lcore \propto Mbh^(0.62\pm0.33) and Lcore \propto L[O III]^(0.79\pm0.30). Above Mbh ~ 10^(6.5) Msun, accretion-driven processes dominate with Lcore \propto Mbh^(1.5-1.65) and Lcore \propto L[O III]^(0.99-1.31) for active galaxies: radio-quiet/loud LINERs, Seyferts, and jetted H II galaxies always display (although low) signatures of radio-emitting BH activity, with L1.5GHz > 10^(19.8) W Hz^(-1) and Mbh > 10^7 Msun, on a broad range of Eddington-scaled accretion rates (mdot). Radio-quiet and radio-loud LINERs are powered by low-mdot discs launching sub-relativistic and relativistic jets, respectively. Low-power slow jets and disc/corona winds from moderately high to high-mdot discs account for the compact and edge-brightened jets of Seyferts, respectively. Jetted H II galaxies may host weakly active BHs. Fuel-starved BHs and recurrent activity account for ALG properties. In conclusion, specific accretion-ejection states of active BHs determine the radio production and the optical classification of local active galaxies.