What determines the nuclear radio emission in local galaxies? 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 HII and Absorption line galaxies [ALG]) galaxies. Using [O III] luminosity ($L_{rm [O~III]}$) 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 ($L_{rm core}$) is found to scale with BH mass ($M_{rm BH}$) and [O~III] luminosity. Below $M_{rm BH} sim$10$^{6.5}$ M$_{odot}$, stellar processes from non-jetted HII galaxies dominate with $L_{rm core} propto M_{rm BH}^{0.61pm0.33}$ and $L_{rm core} propto L_{rm [O~III]}^{0.79pm0.30}$. Above $M_{rm BH} sim$10$^{6.5}$ M$_{odot}$, accretion-driven processes dominate with $L_{rm core} propto M_{rm BH}^{1.5-1.65}$ and $L_{rm core} propto L_{rm [O~III]}^{0.99-1.31}$ for active galaxies: radio-quiet/loud LINERs, Seyferts and jetted HII 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 $M_{rm BH}gtrsim10^{7}$ M$_{odot}$, 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 HII galaxies may host weakly active BHs. Fuel-starved BHs and recurrent activity account for ALG properties. [abridged]
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