DISCERNING THE PHYSICAL ORIGINS OF COSMOLOGICAL GAMMA-RAY BURSTS BASED ON MULTIPLE OBSERVATIONAL CRITERIA: THE CASES OF<i>z</i>= 6.7 GRB 080913,<i>z</i>= 8.2 GRB 090423, AND SOME SHORT/HARD GRBs

(Abridged) The two high-redshift gamma-ray bursts, GRB 080913 at z=6.7 and GRB 090423 at z=8.3, recently detected by Swift appear as intrinsically short, hard GRBs. They could have been recognized by BATSE as short/hard GRBs should they have occurred at z <= 1. We perform a more thorough investigation on two physically distinct types (Type I/II) of cosmological GRBs and their observational characteristics. We reiterate the definitions of Type I/II GRBs and review the observational criteria and their physical motivations. Contrary to the traditional approach of assigning the physical category based on the gamma-ray properties (duration, hardness, and spectral lag), we take an alternative approach to define the Type I and Type II Gold Samples using several criteria that are more directly related to the GRB progenitors, and study the properties of the two Gold Samples and compare them with the traditional long/soft and short/hard samples. We find that the Type II Gold Sample reasonably tracks the long/soft population, although it includes several intrinsically short (shorter than 1s in the rest frame) GRBs. The Type I Gold Sample only has 5 GRBs, 4 of which are not strictly short but have extended emission. Other short/hard GRBs detected in the Swift era represent the BATSE short/hard sample well, but it is unclear whether all of them belong to Type I. We suggest that some (probably even most) high-luminosity short/hard GRBs instead belong to Type II. We suggest that GRB 080913 and GRB 090423 are more likely Type II events. We re-emphasize the importance of invoking multiple observational criteria, and cautiously propose an operational procedure to infer the physical origin of a given GRB with available multiple observational criteria, with various caveats laid out.