We perform a comprehensive analysis of the dynamical magnetic susceptibility $\ensuremath{\chi}(\mathbf{q},\ensuremath{\omega})$ in the slave-boson mean-field scheme of the bilayer $t$-$J$ model. We use model parameters appropriate for ${\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{6+x}$ (YBCO), a typical bilayer high-${T}_{c}$ cuprate compound well studied by neutron scattering experiments. In the $d$-wave pairing state, the strongest magnetic spectral weight appears at $\mathbf{q}=\mathbf{Q}\ensuremath{\equiv}(\ensuremath{\pi},\ensuremath{\pi})$ and $\ensuremath{\omega}={\ensuremath{\omega}}_{\mathbf{Q}}^{\mathrm{res}}$, and spreads into a diamond-shaped shell around $\mathbf{Q}$ in $\mathbf{q}$ space for $\ensuremath{\omega}<{\ensuremath{\omega}}_{\mathbf{Q}}^{\mathrm{res}}$. This weight is due to a collective mode, namely a particle-hole bound state, which has a downward $\ensuremath{\omega}$ versus $\mathbf{q}$ dispersion around $\mathbf{Q}$. Within the high intensity shell, the incommensurate (IC) signals at $\mathbf{q}=(\ensuremath{\pi},\ensuremath{\pi}\ifmmode\pm\else\textpm\fi{}2\ensuremath{\pi}\ensuremath{\eta})$ and $(\ensuremath{\pi}\ifmmode\pm\else\textpm\fi{}2\ensuremath{\pi}\ensuremath{\eta},\ensuremath{\pi})$ tend to be stronger than the diagonal incommensurate (DIC) signals at $\mathbf{q}=(\ensuremath{\pi}\ifmmode\pm\else\textpm\fi{}2\ensuremath{\pi}{\ensuremath{\eta}}^{\ensuremath{'}},\ensuremath{\pi}\ifmmode\pm\else\textpm\fi{}2\ensuremath{\pi}{\ensuremath{\eta}}^{\ensuremath{'}})$, especially for a large hole density $\ensuremath{\delta}$. For $\ensuremath{\omega}\ensuremath{\ll}{\ensuremath{\omega}}_{\mathbf{Q}}^{\mathrm{res}}$ the IC signals completely disappear and the weight remains only around the DIC positions. For $\ensuremath{\omega}>{\ensuremath{\omega}}_{\mathbf{Q}}^{\mathrm{res}}$ strong signals of $\mathrm{Im}\phantom{\rule{0.2em}{0ex}}\ensuremath{\chi}(\mathbf{q},\ensuremath{\omega})$ tracing an upward dispersion are found and interpreted as an overdamped collective excitation near ${\ensuremath{\omega}}_{\mathbf{Q}}^{\mathrm{res}}$. In the normal state, $\mathrm{Im}\ensuremath{\chi}(\mathbf{q},\ensuremath{\omega})$ has a broad peak at $\mathbf{q}=\mathbf{Q}$. That is, the IC and DIC signals appear only in the $d$-wave pairing state. We also study effects of a small orthorhombic anisotropy, which is intrinsic in untwinned YBCO crystals. Because of electron-electron correlations favoring $d$-wave shaped Fermi surface deformations, we expect an enhanced anisotropy of magnetic excitation spectra. This effect is particularly pronounced for low $\ensuremath{\delta}$ and at relatively high temperature. The present theory provides a rather detailed microscopic explanation of the most salient properties of magnetic excitations observed in YBCO.