  We show the existence of two families of short GRBs, both originating from
the merger of binary neutron stars (NSs): family-1 with $E_{iso}<10^{52}$ erg,
leading to a massive NS as the merged core, and family-2 with $E_{iso}>10^{52}$
erg, leading to a black hole (BH). Following the identification of the
prototype GRB 090227B, we present the details of a new example of family-2
short burst: GRB 140619B. From the spectral analysis of the early $\sim0.2$ s,
we infer an observed temperature $kT =(324\pm33)$ keV of the $e^+e^-$-plasma at
transparency (P-GRB), a theoretically derived redshift $z=2.67\pm0.37$, a total
burst energy $E^{tot}_{e^+e^-}=(6.03\pm0.79)\times10^{52}$ erg, a rest-frame
peak energy $E_{p,i}=4.7$ MeV, and a baryon load
$B=(5.52\pm0.73)\times10^{-5}$. We also estimate the corresponding emission of
gravitational waves. Two additional examples of family-2 short bursts are
identified: GRB 081024B and GRB 090510, remarkable for its well determined
cosmological distance. We show that marked differences exist in the nature of
the afterglows of these two families of short bursts: family-2 bursts, leading
to BH formation, consistently exhibit high energy emission following the P-GRB
emission; family-1 bursts, leading to the formation of a massive NS, should
never exhibit high energy emission. We also show that both the families fulfill
an $E_{p,i}$--$E_{iso}$ relation with slope $\gamma=0.59\pm0.07$ and a
normalization constant incompatible with the one for long GRBs. The observed
rate of such family-2 events is
$\rho_0=\left(2.1^{+2.8}_{-1.4}\right)\times10^{-4}$Gpc$^{-3}$yr$^{-1}$.
