Poisson equation in Cosmology at first order in perturbation theory

The book Cosmology by Daniel Baumann states that the Poisson equation for a universe where we consider the effects of both gravity and expansion, expressed in physical coordinates $$\vec{r}=a\vec{x}$$, is the following:

$$\nabla^2\phi=4\pi Ga^2\rho$$

where $$\phi$$ is the gravitational potential, $$a$$ is the scale factor and $$\rho$$ is the density field of the fluid that fills the universe. If we now consider perturbation theory and write:

$$\phi=\bar{\phi}+\delta\phi\ \ \ \ \text{and}\ \ \ \ \rho=\bar{\rho}(1+\delta)$$

where $$\delta\phi$$ is the perturbation in the gravitational potential and $$\delta$$ is the density contrast, then it is stated multiple times in the book that the Poisson equation becomes:

$$\nabla^2\phi=4\pi Ga^2\bar{\rho}\delta$$

However, this is not what I get when I do the calculations. My reasoning is the following:

$$\nabla^2\phi=4\pi Ga^2\rho\ \ \Rightarrow\ \ \nabla^2(\bar{\phi}+\delta\phi)=4\pi Ga^2\bar{\rho}(1+\delta)\ \ \Rightarrow\ \ \nabla^2\bar{\phi}+\nabla^2\delta\phi=4\pi Ga^2\bar{\rho}+4\pi Ga^2\bar{\rho}\delta$$

The only way to get what the book states would be to set $$4\pi Ga^2\bar{\rho}=0$$, which is impossible, as it would mean that $$\bar{\rho}=0$$, and the background density is not zero. And if we consider $$\nabla^2\bar{\phi}=4\pi Ga^2\bar{\rho}$$ (I am not sure whether this is true or not), then what we get is $$\nabla^2\delta\bar{\phi}=4\pi Ga^2\bar{\rho}\delta$$, which is not quite the desired expression either.

Any help would be greatly appreciated!

The definition of $$\phi$$ has been changed, so that it sources peculiar gravitational forces rather than the total gravitational force. That is:
$$\nabla^2\phi=4\pi Ga^2\bar{\rho}\delta$$
This potential sources peculiar velocities $$\vec u$$, such that $$\dot{\vec{u}} = -\nabla\phi$$.
$$\nabla^2\phi=4\pi Ga^2\rho$$
This potential sources physical velocities $$\vec v=H\vec r+\vec u$$, so that $$\dot{\vec{v}} = -\nabla\phi$$. If you split this into 0th and 1st order sets of equations, you should find that the 0th order equations determine the cosmic expansion (at least in a matter-dominated universe) while the 1st order equations govern the perturbations. Cosmological perturbation theory is only concerned with the 1st order equations.