First, let's recall that already in classical theory, Hamiltonian value is not necessarily energy value. Energy is usually defined by some definite expression motivated by a conservation law, which is satisfied in a specific important scenario (e.g. no external forces), but need not be obeyed in the externally driven motion considered. E.g. energy of an harmonic oscillator interacting with a source of external force $F(t)$ is defined as sum of its kinetic and potential energy $\frac{1}{2}mv^2 + \frac{1}{2}kx^2$, so that its value matches value of the standard autonomous Hamiltonian $\frac{p^2}{2m} + \frac{1}{2}kx^2$, not some other valid Hamiltonian giving the same harmonic motion, or value of the full Hamiltonian $\frac{p^2}{2m} + \frac{1}{2}kx^2 - F(t)x $ that describes the externally driven motion.
In quantum theory, different choices of the Hamiltonian may produce different eigenvalues, so the Hamiltonian $H_{system} + H_{interaction}$, in general, has different eigenvalues than the Hamiltonian $H_{system}$ has.
However, similarly to classical theory, energy levels of a system are defined by eigenvalues of a particular Hamiltonian chosen for the purpose, often following the classical convention that energy is a sum of kinetic and potential energy, excluding the terms due to external forces(fields). In quantum theory, energy is not always defined for all states, but definition of energy eigenvalues follows this rule.
If the term due to external forces depends only very slowly on time, both in classical and quantum theory, we can consider it part of definition of energy (energy eigenvalues).
One convention, used e.g. in description of atom-light interaction, is as follows. If the full Hamiltonian that defines evolution of the psi function has the form
$$
H_0 + H_p(t),
$$
where $H_0$ does not depend on time, and $H_p$ depends on time, but slowly enough (so that evolution of the psi function is "adiabatic", that is, with eigenfunctions changing in time, while magnitudes of the expansion coefficients remaining approximately constant), then $H_p$ is considered to be a part of the Hamiltonian for the purpose of defining eigenfunctions and eigenvalues.
That is quite a special condition, which holds if $H_p$ time dependence is due to some slowly varied macroscopic parameter, such as external electrostatic or magnetostatic field. The observed frequencies in the absorption spectrum then are close to differences of such time-dependent eigenvalues of $H_0 + H_p$ (e.g. the Stark effect, or the Zeeman effect).
The above does not apply when the electric or magnetic field oscillates rapidly, comparably to transition frequencies of the atom; then the observed frequencies in the absorption spectrum are close to differences of the time-independent eigenvalues of $H_0$. So if the condition does not hold, it is better to define the eigenvalues based on the time-independent Hamiltonian $H_0$.
In quantum statistical physics of macroscopic systems, the situation is somewhat different, due to large number of particles and dominance of the system energy compared to interaction energy, but the result is similar. The interaction term is supposed to be time-independent, so it could be considered part of $H_0$, but it is also "many times smaller" than the system Hamiltonian, in its effect on eigenvalues. Inclusion of this term would make the eigenvalues dependent on coordinates of the bath, which would be a complication with a small effect on the result. Thus we do not include the interaction term in the Hamiltonian which defines the system eigenvalues.
If the interaction term is "large enough" (its inclusion has significant effect on the eigenvalues), like is the case with gas with changing volume, or a ferromagnet with changing external magnetic field, then we include this interaction term in the Hamiltonian $H_0$ that defines system energy eigenvalues.
