= Solution
Put $\rho=\varphi^\dagger\varphi$. The heavy-field <equation of motion> is
$$
(\Box+M^2)\phi=-g\rho,
\qquad
\phi=-\frac g{M^2}\rho+\frac g{M^4}\Box\rho+O(M^{-6}).
$$
This solution describes the low-energy sector without an independent incoming heavy-particle wave. It is valid when all external <momentum> invariants and light masses squared are small compared with $M^2$.
After an integration by parts, the terms involving the heavy <field> are $-\tfrac12\phi(\Box+M^2)\phi-g\rho\phi$. Substituting the solution into both terms, rather than the source term alone, gives
$$
\Delta\mathcal L_{\rm eff}=\frac{g^2}{2}\rho(\Box+M^2)^{-1}\rho
=\frac{g^2}{2M^2}\rho^2-\frac{g^2}{2M^4}\rho\Box\rho+O(M^{-6}).
$$
Hence, with the question's sign convention for the local <quartic interaction>,
$$
\boxed{\lambda=-\frac{g^2}{2M^2},\qquad
\mathcal L_{\rm eff}=\partial_\mu\varphi^\dagger\partial^\mu\varphi-m^2\rho-\lambda\rho^2+O(M^{-4}).}
$$
This is <integrating out a field> followed by a <derivative expansion>. The negative quartic means attraction. It also makes the truncated potential unbounded at large fields; minimizing the original potential over $\phi$ reveals the same negative quartic. Thus neither displayed model, without extra stabilizing interactions, supplies a globally stable nonperturbative vacuum. The matching describes perturbative low-energy scattering about the local vacuum. The simple equation-of-motion matching fixes the leading tree-level coefficient; precision quantum matching requires consistent loop renormalization and the additional effective operators at the requested accuracy.
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