We analyzed the initial rising behaviors of X-ray outbursts from two transient low-mass X-ray binaries (LMXBs) containing a neutron-star (NS), Aquila X-1 (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 can be classified into two types based on 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 15–50 keV intensities achieve the maximums after the transition. The former have the longer initial hard-state ($ \gtrsim$ 9 d) than the latter ($ \lesssim$ 5 d). Therefore, we named them as slow-type (S-type) and fast-type (F-type), respectively. These two types also show 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 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 $ \times$ 10$ ^{36}$ erg s$ ^{-1}$ in the S-type, which corresponds to 4% of the Eddington luminosity for a 1.4 NS.