= Solution
Both descriptions are depth-averaged <Hydrostatic equilibrium>[hydrostatic] shallow-layer models. In a one-layer free-surface <shallow water equations>, a homogeneous layer overlies a rigid bottom and its upper surface moves. Surface elevation changes give a restoring acceleration based on full gravity $g$, with external-wave speed $\sqrt{gH_0}$. <Barotropic mode>[Barotropic] coastal storm surges, long surface <Gravity wave>[gravity waves] or a basin seiche are examples.
In the rigid-lid <reduced gravity> model, the movable boundary is a <density> interface beneath the active upper layer, while the much deeper lower layer remains at rest. The internal interface mode instead has speed $\sqrt{g'h_0}$ and represents, for example, thermocline displacements and an upper-ocean <Baroclinic mode>[baroclinic eddy]. The mathematical shallow-water form is retained with $g$ replaced by $g'$ and with the appropriate interface-depth convention. A freely moving two-layer surface would support both external and internal modes; the rigid lid filters out the fast external mode, and the resting abyss removes its independent dynamics.
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