We present the most extensive sample of 45 type I (short) and 275 type II (long) gamma-ray bursts (GRBs) with known redshift to investigate the correlation between the rest-frame peak energy, E_p,i_ and the total isotropic equivalent energy, E_iso_ of the prompt emission (Amati relation). The E_p,i_-E_iso_ correlation for type I bursts is found to be well distinguished from the one constructed for type II bursts and has a similar power-law index value, E_p,i_{prop.to}E_iso_^0.4^, which possibly indicates the same emission mechanism of both GRB types. We show that the initial pulse complex (IPC) of type I bursts with an extended emission and regular type I bursts follow the same correlation. We obtain similar results for type II bursts associated with Ic supernovae and for regular type II bursts. Three possible outliers from the E_p,i_-E_iso_ correlation for type II subsample are detected. Significant evolution of the E_p,i_-E_iso_ correlation with redshift for type II bursts is not found. We suggest the new classification method, based on the E_p,i_-E_iso_ correlation and introduce two parameters, EH=E_p,i,2_E_iso,51_^-0.4^ and EHD=E_p,i,2_ E_iso,51_^-0.4^T_90,i_^-0.5^, where E_p,i,2_ is the value of E_p,i_ parameter in units of 100keV, E_iso,51_ is the value of E_iso_ parameter in units of 10^51^erg, and T_90,i_ is the rest-frame duration in units of seconds. EHD is found to be the most reliable parameter for the blind type I/type II classification, which can be used to classify GRBs with no redshift.