  We analyzed the initial rising behaviors of X-ray outbursts from two
transient low-mass X-ray binaries (LMXBs) containing a neutron-star (NS), Aql
X-1 and 4U 1608-52, which are continuously being monitored by MAXI/GSC in 2--20
keV, RXTE/ASM in 2--10 keV, and Swift/BAT in 15--50 keV. We found that the
observed ten outbursts are classified into two types by the patterns of the
relative intensity evolutions in the two energy bands below/above 15 keV. One
type behaves as the 15--50 keV intensity achieves the maximum during the
initial hard-state period and drops greatly at the hard-to-soft state
transition. On the other hand, the other type does as both the 2--15 keV and
the 15--50 keV intensities achieve the maximums after the transition. The
former have the longer initial hard-state ($\gtrsim$ 9 d) than the latter's
($\ltsim$5 d). Therefore, we named them as slow-type (S-type) and fast-type
(F-type), respectively. These two types also show the differences in the
luminosity at the hard-to-soft state transition as well as in the average
luminosity before the outburst started, where the S-type are higher than the
F-type in the both. These results suggest that the X-ray radiation during the
pre-outburst period, which heats up the accretion disk and delays the disk
transition (i.e., from a geometrically thick disk to a thin one), would
determine whether the following outburst becomes S-type or F-type. The
luminosity when the hard-to-soft state transition occurs is higher than $\sim 8
\times10^{36}$ erg s$^{-1}$ in the S-type, which corresponds to 4% of the
Eddington luminosity for a 1.4 \Mo NS.
