Neutron experiments have indicated that the structure factor S (q, co) for the two cuprates YBa2Cu307 q and La~Sr Cu04 has a different q dependence. Commensurate peaks at (~/a, m/a) are observed in the former case, whereas clearly incommensurate peaks are seen in the latter, for metallic hole concentrations. We attribute this contrasting q dependence to differences in the Fermi-surface geometry, obtained in band-structure approaches, and (for the YBaCuO system) also corroborated by photoemission experiments. Using a large Coulomb-U, Fermi-liquid-based scheme, we present results for the q, co and temperature dependence of the neutron cross section as well as for the temperature dependence of the NMR relaxation, in both cuprate families at various hole concentrations. When antiferromagnetic quasiparticle interactions of moderate strength are included, these calculations compare favorably with experiment.It should be stressed that the Fermi-surface shape must be accurately represented in both systems in order to produce this good agreement with the neutron data. We conclude that the close correspondence found, thus far, between band-structure-derived spin dynamics and the detailed fermiology of both cuprates provides support for Fermi-liquid-based schemes. Furthermore, this correspondence suggests important constraints which should be included in theoretical schemes ranging from the marginal and nearly antiferromagnetic Fermi liquid to more exotic scenarios for the normal state. Within this context, it is extremely important to determine the characteristic energy scales of the Fermi liquid. Comparison of our calculations with the measured energy scales of the spin dynamics indicates that these are sufficiently low so that one can reconcile deviations from canonical behavior above T, with a Fermi-liquid ground state. Explicit effects of these low-energy scales are discussed in the context of the quasiparticle lifetime as a function of frequency and temperature. Our detailed studies also yield predictions for future experiments which will help to test futher the validity of this approach.