This is a very nice question touching ongoing research. I cannot really provide a nice answer to it but I would like to share a few thoughts I had when thinking about this question.
1. I think mentioning the Hohenberg-Kohn theorem in the question puts Becke's statement slightly into the wrong context. In this context one should first realize that there is an object even simpler than the ground-state electron density containing all details of the investigated system: The many-body Hamiltonian
$$\hat{H} = \hat{T} + \hat{V}_\text{ee} + \hat{V}_\text{ext}$$
consisting of the kinetic energy operator $\hat{T}$, the Coulomb interactions $\hat{V}_\text{ee}$, and the external potential $\hat{V}_\text{ext}$. You can actually write it down in every detail on a sheet of paper.
Hohenberg and Kohn show for Hamiltonians of this form with the known expressions for $\hat{T}$ and $\hat{V}_\text{ee}$ that there is a one-to-one mapping between the ground-state density and the external potential. At least up to a constant potential shift.
Essentially this means that the positions and charges of the atomic nuclei are encoded in the ground-state density. But more complex external potentials are also possible. The form of the external potential does not have to be known beforehand. For the rest of the Hamiltonian this is not the case. It is assumed that it has the form sketched above: There is no one-to-one mapping between the density and the form of $\hat{T}$ and $\hat{V}_\text{ee}$ which actually contain all the interactions you asked about.
There even is an obvious counterexample for such a one-to one mapping: Kohn and Sham construct an auxiliary system of noninteracting electrons featuring the same ground-state density.
So on this level the ground-state density only in connection with knowledge of the form of $\hat{T}$ and $\hat{V}_\text{ee}$ contains all the interesting interaction details. But as mentioned this is also encoded in the Hamiltonian.
2. I think Becke's statement has to be seen in the context of energy functionals to extract the ground-state energy of an interacting many-electron system from the respective density.
Such energy functionals are typically based on the Kohn-Sham system which is constructed such that the most significant contributions to the energy of the interacting many-electron system are easily accessible. This includes the Hartree energy, the energy due to the external potential, and the kinetic energies of the single-electron Kohn-Sham orbitals. Everything beyond these energy contributions is integrated into the exchange-correlation energy for which no exact expression is known.
Fortunately this energy contribution often is not the dominating part and even simple approximations cover it with a good enough accuracy, e.g., the local density approximation that assumes the exchange-correlation energy of a homogeneous electron gas. For the homogeneous electron gas an expression for the exchange energy in terms of the density is known and the correlation energy can numerically be calculated by simulations.
As indicated in my first sentence of this answer improving the accuracy of approximations to the exchange-correlation functional is ongoing research. One knows several properties the exact exchange-correlation functional has to fulfill. I think getting an intuitive physical picture on how the interactions are encoded in the ground-state density is strongly connected to knowing and understanding these properties. Unfortunately I cannot give you insights on this.
Systematic improvements to the exchange-correlation functional often go beyond only using the charge density in a direct way. Since most of the exchange-correlation energy is due to exchange you see many approaches to express the exchange part exactly in terms of Kohn-Sham orbitals. This can then be used to construct a hybrid functional or to combine it with the random phase approximation to the correlation energy.
Of course, one actually does not want to use the Kohn Sham orbitals for this because the resulting expressions are connected to significant computational demands. But the fact that it is done in this way indicates how difficult it is to directly extract the exchange-correlation energy from the density. It might also indicate that a practically usable physical picture directly connecting the density to the respecive interactions is not really available. But of course, it might also indicate that the encoding of this quantity in the rather featureless electron density is just too subtile to be usable.
