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 ($L_{\rm [O\, \small {III}]}$) 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 ~ 106.5 M⊙, stellar processes from non-jetted H II galaxies dominate with $L_{\rm core} \propto M_{\rm BH}^{0.61\pm 0.33}$ and $L_{\rm core} \propto L_{\rm [O\, \small {III}]}^{0.79\pm 0.30}$. Above MBH ~ 106.5 M⊙, accretion-driven processes dominate with $L_{\rm core} \propto M_{\rm BH}^{1.5-1.65}$ and $L_{\rm core} \propto L_{\rm [O\, \small {III}]}^{0.99-1.31}$ for active galaxies: radio-quiet/loud LINERs, Seyferts, and jetted H II galaxies always display (although low) signatures of radio-emitting BH activity, with $L_{\rm 1.5\, GHz}\gtrsim 10^{19.8}$ W Hz-1 and MBH ≳ 107 M⊙, on a broad range of Eddington-scaled accretion rates ($\dot{m}$). Radio-quiet and radio-loud LINERs are powered by low-$\dot{m}$ discs launching sub-relativistic and relativistic jets, respectively. Low-power slow jets and disc/corona winds from moderately high to high-$\dot{m}$ 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.