The electric field lines denote the electric field intensity $\vec{E}$ at every point so why doesn't magnetic field lines denote the magnetic field intensity $\vec{H}$ at every point but the magnetic flux density $\vec{B}$?

I understand that basically it is the same thing and it actually the lines do denote $\vec{H}$ as well because:

$$\vec{B}=\mu \vec{H}$$

but I haven't seen a single source mentioning that the magnetic field lines express $\vec{H}$.

Also the electric flux is is given by:

$$\Phi = \iint\_{S}\vec{E}\cdot d\vec{S}$$

So why isn't the magnetic flux given by:

$$\Phi = \iint\_{S}\vec{H}\cdot d\vec{S}$$

Again it is the same thing and it is only a matter of semantics but I have yet to come across a source using the latter expression for the magnetic flux.