  We provide a unified interpretation of both paramagnon and plasmon modes in
high-$T_c$ copper-oxides, and verify it quantitatively against available
resonant inelastic $x$-ray scattering (RIXS) data across the hole-doped phase
diagram. Three-dimensional extended Hubbard model, with included long-range
Coulomb interactions and doping-independent microscopic parameters for both
classes of quantum fluctuations, is used. Collective modes are studied using
VWF+$1/\mathcal{N}_f$ approach which extends variational wave function (VWF)
scheme by means of an expansion in inverse number of fermionic flavors
($1/\mathcal{N}_f$). We show that intense paramagnons persist along the
anti-nodal line from the underdoped to overdoped regime and undergo rapid
overdamping in the nodal direction. Plasmons exhibit a three-dimensional
character, with minimal energy corresponding to anti-phase oscillations on
neighboring $\mathrm{CuO_2}$ planes. The theoretical spin- and charge
excitation energies reproduce semi-quantitatively RIXS data for $\mathrm{(Bi,
Pb)_2 (Sr, La)_2 CuO_{6+\delta}}$. The present VWF+$1/\mathcal{N}_f$ analysis
of dynamics and former VWF results for static quantities combine into a
consistent description of the principal properties of hole-doped high-$T_c$
cuprates as strongly correlated systems.
