  We present a combined oxygen $K$-egde x-ray absorption spectroscopy (XAS) and
resonant inelastic x-ray scattering (RIXS) study of the bilayer ruthenate
Ca$_3$Ru$_2$O$_7$. Our RIXS experiments on Ca$_3$Ru$_2$O$_7$ were carried out
on the overlapping in-plane and inner apical oxygen resonances, which are
distinguishable from the outer apical one. Comparison to equivalent oxygen
$K$-edge spectra recorded on band-Mott insulating Ca$_2$RuO$_4$ is made. In
contrast to Ca$_2$RuO$_4$ spectra, which contain excitations linked to Mott
physics, Ca$_3$Ru$_2$O$_7$ spectra feature only intra-$t_{2g}$ ones that do not
directly involve the Coulomb energy scale. As found in Ca$_2$RuO$_4$, we
resolve two intra-$t_{2g}$ excitations in Ca$_3$Ru$_2$O$_7$. Moreover, the
lowest lying excitation in Ca$_3$Ru$_2$O$_7$ shows a significant dispersion,
revealing a collective character differently from what is observed in
Ca$_2$RuO$_4$. Theoretical modelling supports the interpretation of this lowest
energy excitation in Ca$_3$Ru$_2$O$_7$ as a magnetic transverse mode with
multi-particle character, whereas the corresponding excitation in Ca$_2$RuO$_4$
is assigned to combined longitudinal and transverse spin modes. These
fundamental differences are discussed in terms of the inequivalent magnetic
ground-state manifestations in Ca$_2$RuO$_4$ and Ca$_3$Ru$_2$O$_7$.
