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 ($L[O III]$) 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 $M_{\rm BH} \sim$10$^{6.5}$ M$\mathrm{<mrow\; data-mjx-texclass="ORD">\odot </mrow>}$, stellar processes from non-jetted HII galaxies dominate with $Lcore\propto MBH0.61\pm 0.33$ and $Lcore\propto L[O III]0.79\pm 0.30$. Above $M_{\rm BH} \sim$10$^{6.5}$ M$\mathrm{<mrow\; data-mjx-texclass="ORD">\odot </mrow>}$, accretion-driven processes dominate with $Lcore\propto MBH1.5-1.65$ and $Lcore\propto L[O III]0.99-1.31$ 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$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$ and $MBH\gtrsim 107$ M$\mathrm{<mrow\; data-mjx-texclass="ORD">\odot </mrow>}$, on a broad range of Eddington-scaled accretion rates ($m\dot{}$). Radio-quiet and radio-loud LINERs are powered by low-$m\dot{}$ discs launching sub-relativistic and relativistic jets, respectively. Low-power slow jets and disc/corona winds from moderately high to high-$m\dot{}$ 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]