Most books written in the last 100 years would say that $B$ is the true or real physical field an $H$ is secondary and derived from $B$, while before and during the development of EM reverse view was prevalent that $H$ be the primary; engineers still prefer to start with $H$ for very practical reasons.
Since in a vacuum the two fields, $H$ and $B=\mu_0 H$ are the same except for a dimensional coefficient $\mu_0$ whose value is determined by convention it makes no sense to assign meaning to the question of which quantity is more fundamental. In a macroscopic magnetic material the situation is different. The two fields have different behaviors across boundaries: specifically, $B_n$ is always continuous while $\Delta H_t=J_t$ representing the tangential jump in the $H$ field equals the surface current flowing between the two media; if $J_t=0$, no surface current, then $\Delta H_t=0$, in other words, $H_t$ is continuous across the boundary.
Because of these differences their measurements are different inside magnetic matter. We usually associate force with $H$ and torque with $B$ but to measure these you need an empty cavity within the magnetic body. Due to the boundary condition differences one employs different shaped cavities, Kelvin's crevices. To measure the force a long and narrow needle-shaped cavity that is parallel with the local $H$ field inside the body will do because in the needle cavity and just outside of it but still within the magnetic body the two the $H$ fields will be equal. Contrariwise, a disk-shaped flat and thin cavity perpendicular to the local $B$ field will have the same $B$ field inside as it is immediately outside. These differences can be shown to be captured by the introduction of the magnetic polarization density field denoted by $M$, so that $B=\mu_0(H+M)$. It is completely without physical significance with what sign you define this $M$, one could just as well have defined it as $B=\mu_0 (H-M).$
There are two "causes" for a magnetic field in classical EM, one cause is a moving charge, a current, and the other possible cause is an irreducible magnetic dipole as the source. Neither can be explained any more than the cause for inertia in Newtonian mechanics. It just is. Since a circular current has the same magnetic field at distances large relative to its diameter it can be assigned a magnetic dipole moment similar to that of the irreducible magnetic moments, the macroscopic averages of fields of particles and atomic/electronic currents can be represented by $M$. Note that the relationship among these three macroscopic fields $f(B,H,M)=0$ is no more accessible to classical analysis than, say, the ideal gas law $pV=nRT$ or Hooke's law $F=-kx$. These are macroscopic constitutive "laws" defining the body under examination. Their explanation is outside of EM, mechanics, and thermodynamics; they are assumed subjects of investigation but not explained. Their explanation is in statistical mechanics, solid-state physics, quantum mechanics, etc.
