= Solution
Use small <Rossby number> $U/(f_0l)\ll1$, slow evolution on the advective time scale, small interface displacements relative to each layer depth, shallow hydrostatic layers, stable <reduced gravity> $g'>0$, and inviscid unforced flow. On a <beta plane>, take $\beta l/f_0$ of the same small order as the <Rossby number>. The internal <Burger number> is retained at order unity so stratification and relative-vorticity effects can both enter the leading potential-vorticity anomaly.
The <rigid-lid pressure in two-layer flow> comes from neglecting the free-surface volume displacement in the <rigid-lid approximation>; it does not permit setting the common horizontal pressure gradient to zero. The small surface displacement multiplied by $g$ retains a finite lid-pressure multiplier. Write $h_1=H_1+\chi$, $h_2=H_2-\chi$, and let $\Pi$ denote this common pressure potential. Leading <geostrophic balance> gives
$$
f_0\psi_1=\Pi,\qquad f_0\psi_2=\Pi-g'\chi,\qquad
\boxed{\chi=\frac{f_0}{g'}(\psi_1-\psi_2),}
$$
with $\mathbf u_i=(-\psi_{iy},\psi_{ix})$ at leading order. Literally setting $h_1+h_2$ to a spatial constant in the momentum gradients before taking the rigid-lid limit would suppress the upper-layer pressure field and fail to produce general two-layer QG dynamics.
Taking curl of each shallow-water momentum equation and using layer continuity gives material conservation of $(f+\zeta_i)/h_i$. Expanding it to first order and advecting the anomaly by the leading geostrophic velocity yields
$$
\boxed{q_1=\nabla_h^2\psi_1+F_1(\psi_2-\psi_1)+\beta y,\qquad
q_2=\nabla_h^2\psi_2+F_2(\psi_1-\psi_2)+\beta y,\qquad
F_i=\frac{f_0^2}{g'H_i}.}
$$
The <two-layer quasi-geostrophic potential vorticity> equations are
$$
\boxed{\partial_tq_i+J(\psi_i,q_i)=0,\qquad
J(A,B)=A_xB_y-A_yB_x,\quad i=1,2.}
$$
Here the common background $f_0/H_i$ has been removed and the anomaly multiplied by $H_i$. The advection term is retained at the same slow order as the time derivative even though the leading velocity/pressure balance was linear geostrophy. On an $f$ plane simply set $\beta=0$.
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