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Based on the analysis of GRB 090227B, the prototype of short gamma-ray bursts (SGRBs) originating in binary neutron star (NS) mergers leading to black hole (BH) formation, the progress on the determination of NS critical mass, and the information gained from binary NS systems in our own Galaxy, we proposed the existence of two families of SGRBs originating from NS mergers: family-1 SGRBs with isotropic energy Eiso<10^52 erg and rest-frame spectral peak energy Ep<2 MeV, leading to a massive NS as the merged core; family-2 SGRBs with Eiso>10^52 erg and Ep>2 MeV, leading to a BH as merged core. After GRB 090227B, we present the family-2 SGRB 140619B. From the spectral analysis of the early 0.2 s, we infer an observed temperature $kT =(324\pm33)$ keV of the $e^+e^-$-plasma at transparency (P-GRB), a theoretically derived redshift z=2.67, a total burst energy E_iso=6.03x10^52 erg, a rest-frame peak energy E_p=4.7 MeV, and a baryon load B=5.52x10^(-5). The analysis of the prompt emission led to the determination of a circum-burst medium density nCBM=4.7x10^(-5) cm^(-3), typical of the galactic halo environment, where NS-NS binaries migrate to, due to natal kicks imparted to them at birth. We also estimate the corresponding emission of gravitational waves. Two additional examples of family-2 SGRBs are identified: GRB 081024B and GRB 090510, remarkable for its well determined cosmological distance. Marked differences exist in the afterglows of these two families of short bursts: family-2 SGRBs, leading to BH formation, consistently exhibit high energy emission following the P-GRB emission; family-1 SGRBs, leading to the formation of a massive NS, have extended X-ray afterglow but should never exhibit high energy emission. We also show that both the families fulfill an E_p-E_iso relation with a normalization constant incompatible with the one for long GRBs. The observed rate of such family-2 events is 2.1x10^(-4) Gpc^(-3)yr^(-1).

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