In many electrodynamics/magnetism textbooks it is stated that the magnetic induction $\vec{B}$ is related to the magnetic field $\vec{H}$ via $\vec{B}=\mu_0(\vec{H}+\vec{M})$. It is then argued that $\vec{H}$ is an important quantity because it is given by the free currents only, so that $\nabla \times \vec{H} = \vec{j}_\text{free}$.
Free currents are something we control in experiments and so tuning $\vec{H}$ via $\vec{j}_\text{free}$ is straightforward. In addition, it is often argued that $\vec{H}$ is independent of the medium, so it really makes sense to think about everything in terms of $\vec{H}$. All of this gave me the impression that $\vec{H}$ can be thought of as some sort of an external field we impose on our sample, which then responds through some magnetization $\vec{M}$ and then the net field inside the sample would be $\vec{H}+\vec{M}$. I never really understood why we needed to introduce a new quantity $\vec{H}$, instead of just writing the induction as $\vec{B}_\text{tot} = \vec{B}_\text{ext} + \mu_0\vec{M}$ where $\vec{B}_\text{tot}$ would be the total induction in the sample and $\vec{B}_\text{ext}$ would be the induction imposed externally, that we would measure if the sample was not there.
But then I learned about shape anisotropy and demagnetizing fields which show that the non-uniformities (specifically, divergence) of $\vec{M}$ can also result in an extra $\vec{H}$ field. If I'm correct, $\vec{j}_\text{free}$ controls the curl and $\nabla \cdot\vec{M}$ controls the divergence of $\vec{H}$. However, if that's the case, $\vec{H}$ is really not material independent. What would be a good, intuitive way to think about $\vec{H}$ then? And what is all the fuss about $\vec{H}$ if it's i) material dependent and ii) cannot even be measured or controlled in the material, independently of $\vec{M}$?