  Superconducting (SC) fluctuations in cuprate superconductors have been
extensively studied to gain a deep insight into preformed Cooper pairs above
the SC transition temperature $T_{\text{c}}$. While the various measurements,
such as the terahertz (THz) optical conductivity, Nernst effect, angle-resolved
photoemission spectroscopy (ARPES), and scanning tunneling microscopy (STM)
measurements have provided the signature of the SC fluctuations, the onset
temperature of the SC fluctuations depends on the measurement scheme. Here, we
shed light on the Higgs mode to investigate the SC fluctuations, as it is the
direct fingerprint of SC order parameter and can help elucidate the development
of SC phase coherence. We perform THz pump-optical probe spectroscopy for
underdoped and overdoped Bi$_2$Sr$_2$CaCu$_2$O$_{8+x}$ (Bi2212) thin films. The
oscillatory behavior in the pump-probe signal (THz Kerr signal) observed in the
SC phase has been identified as the Higgs mode in single crystals in our
previous work [K. Katsumi et al., Phys. Rev. Lett. 120, 117001 (2018)], but two
onset temperatures are identified above $T_{\text{c}}$. Combined with the
results of the single crystals in a wide range of doping, we find that the
first onset $T_1^{\text{ons}}$ is 10-30 K above $T_{\text{c}}$.
$T_1^{\text{ons}}$ coincides with that of the superfluid density $N_s$
extracted from the THz optical conductivity. Hence, $T_1^{\text{ons}}$ is
interpreted as the onset of macroscopic SC phase stiffness. On the other hand,
the second onset $T_2^{\text{ons}}$ is identified at substantially higher than
$T_{\text{c}}$, whose origin is discussed in terms of preformed Cooper pairs.
