Conceptual definition of the $H$-field I am looking for a conceptual definition or statement for H to help ground my understanding as I work through the math.
I often see it explained that D and H are just the components of the fields solely due to the free charge and current respectively. But this always struck me as an imprecise way to look at it. Griffiths gave the example of the electret where there is a D even without free charge, not to mention the fact that the units are different than the respective fields suggesting it’s more complicated.
I’ve used the following statement for D to help me conceptualize it:

the charge per unit area that would be displaced across a layer of conductor placed across an electric field. This describes also the charge density on an extended surface that could be causing the field.

Could there be a similar definition for H? Given that the SI units are [A/m] maybe something like the current generated across some length of wire that’s always perpendicular to a magnetic field?
 A: 
I often see it explained that D and H are just the components of the fields solely due to the free charge and current respectively. But this always struck me as an imprecise way to look at it. 

You are correct, it is not generally true. It is true in special situations like infinite plane parallel capacitor with dielectric, or toroidal coil with nonconducting ferrite core.
The best definition that I know, that covers most cases, is this:
$$
\mathbf D = \epsilon_0 \mathbf E + \mathbf P
$$
$$
\mathbf H = \mathbf B/\mu_0 - \mathbf M.
$$
There is no obvious "physical interpretation" of these new quantities than that these quantites make equations look simpler, and that in simple situations, they obey simple rules, such as in the case of plane parallel capacitor or toroidal coil. Analyzing static fields in material cavities, people came to an interesting interpretation that (in material medium) $\mathbf D$ at some point of material medium is (ignoring units) electric field in vacuum that would be present inside a thin oblate cavity if it was cut there, while $\mathbf E$ is electric field for a long prolate cavity. Similarly for $\mathbf B$: this is magnetic field in vacuum that would be present if there was an oblate cavity there, while $\mathbf H$ (ignoring units) is field for a prolate cavity.
Getting over these simple examples, the above definition is general, mathematical and involves understanding four quantities $\mathbf E,\mathbf B,\mathbf P,\mathbf M$. 
Fields $\mathbf E$, $\mathbf B$ are understood as macroscopic EM fields, obeying Maxwell's equations for vacuum
$$
\nabla\cdot\mathbf E = \frac{\rho_{total}}{\epsilon_0}
$$
$$
...
$$
where total charge and total current density contain contribution due to material bodies (in other words, material medium is treated as vacuum with not only free charge and current, but additional electric charge and current contribution due to matter).
The fields $\mathbf P,\mathbf M$ are an approximate way to encapsulate to most important character of those matter contributions to charge and current density. Polarization $\mathbf P$, for nonconducting medium made of electrically neutral clusters, gives average electric dipole moment of clusters associated with a unit volume. Magnetization $\mathbf M$ gives average magnetic moment of similar clusters associated with unit volume.
