There is experimental evidence suggesting the existence of a liquid-liquid transition (LLT) in a single-component liquid. However, none of this evidence is free from controversy, including the case of a molecular liquid, triphenyl phosphite, which we study here. Furthermore, the kinetics of LLT has been largely unexplored. Here we study the phase-transition dynamics of triphenyl phosphite in a supercooled liquid state by means of time-resolved polarized and depolarized small-angle light scattering to clarify whether the transition is a liquid-liquid transition (LLT) or merely nanocrystal formation. A part of this study was recently reported in another of our papers [Shimizu, R.; Kobayashi, M.; Tanaka, H. Phys. Rev. Lett. 2014, 112, 125702]. A detailed analysis of our experimental results of light scattering and the comparison with heat evolution during LLT have revealed the following facts. The polarized scattering from domains has a finite (nonzero) intensity in the low-wavenumber limit, and the time evolution of its average intensity is almost proportional to the square of the heat-releasing rate. The depolarized scattering intensity monotonically increases in the process of LLT during isothermal annealing above the spinodal temperature T-SD but exhibits a peak below T-SD. On the basis of these results, we suggest that the primary process is LLT, whose order parameter is of a nonconserved nature, but accompanies nanocrystal formation. In the NG-type LLT, the sharp interface between liquid II droplets and the liquid I matrix promotes nanocrystal formation there, whereas much less nanocrystal formation is induced in the SD-type LLT due to the lack of such sharp interfaces.