We could use any self-gravitating fluid for this question, but let's take a star as an example. Left alone, it's density is the source of it's potential and thus we have the simplest form of Poisson's equation:
$$ 4 \pi G \rho_0 = \nabla^2 \Phi_0 $$
Now let this system be perturbed by an outside potential, $U$ (which we assume is small compared to the star's own potential, $\Phi_0 \gg U$). An example physics case would be another star passing close by, and thus $U$ is approximately a point mass potential.
Let's take the first order perturbations to both the potential and the density: $$\rho = \rho_0 + \rho' + O(2)$$ and $$\Phi = \Phi_0 + \Phi' + U + O(2)$$
My question boils down to this, which of the following is the correct formulation of the first order Poisson equation? :
$$ 4 \pi G \rho' = \nabla^2 \Phi' $$ OR $$ 4 \pi G \rho' = \nabla^2 (\Phi' + U) $$
I'm inclined to say the first is correct, as it makes sense to me that the relationship only applies to quantities describing the self-gravitating fluid.
However, the perturbed density is certainly affected by the external potential. I imagine this effect could be written implicitly into the perturbed potential, but I'd be very interested to hear the thoughts of the community on this fine-detail point.