You are not the first to be befuddled by conventional explanations of B & H as they apply to practical electromagnetic devices such as ferrite inductor cores. I struggled for years with the standard explanations of the nature of B & H and their application in such devices. My salvation came from a single chapter in a largely forgotten book I happened upon in a used book store some twenty-odd years ago. I believe the book is now available on-line in pdf format. Try Google Books. The name of the book is "The Magnetic Circuit" by V. Karapetoff and was published around 1911 - yes, 110+ years ago! Nonetheless, magnetic principles were well understood at the time and the terminology has been essentially unchanged in the intervening decades.
If you read Chapter 1 very carefully you will be blessed with a very practical understanding of the magnetic field and all of its beautiful characteristics and its arcane terminology which is still in common use today (e.g. magnetomotive force, permeance, reluctance, flux vs flux density, etc.) The remaining chapters are also interesting, but not as well presented as Chapter 1, which I revere as a sparkling gem of engineering exposition.
It will also help your understanding if you construct a few simple air-core coils to experiment with as an aid to digestion of the basic concepts. Use a function generator to drive the coils and a smaller coil to sense the magnetic field and display it on an oscilloscope. The driven coils should be about 6-12 inches in diameter and the sense coil about 1/2" in diameter. A frequency of 1000 Hz is adequate. If you are really ambitious you should build the toroidal coil which the author uses as his main vehicle of explanation.
I'll end by giving my standard explanation of B & H: The simplest electrical circuit is a battery with a parallel connected resistor. Ohms Law can be learned solely from this simple arrangement of three elements - voltage source, resistance and wire - along with a voltmeter and ammeter. B & H can be analogously learned from the simplest magnetic circuit. This is a wire with a current (AC or DC) flowing through it.
The magnetic field produced by the current encircles the wire with a cylindrical formation of flux lines. "M" is the magnetomotive force analogous to the voltage of the battery in the Ohms Law example. "B" is the strength of the resulting magnetic flux field formed around the wire by that magnetomotive force M, and is analogous to the electrical current "I" in the Ohms Law example. The "resistor" is the permeability of the air surrounding the wire. The surrounding air forms a "collective" or "distributed" magnetic resistor of sorts around the wire. This "magnetic resistor" dictates a ratio of produced flux "B" for a given driving force (i.e. magnetomotive force) "M", which is in turn proportional to the value of the current flowing thru the wire, quite similar to Ohms Law. Unfortunately, we cannot purchase "magnetic resistors" in any value which suits our fancy. Nor is there a "Magnetomotive Force Meter" equivalent to our handy voltmeter available from Digikey. If you are fortunate enough to have a "flux meter" you could measure the "B" value of the flux lines surrounding the wire. So, imagine how you would decipher Ohms Law from the simple battery-resistor circuit I described above, if all you had to work with was an ammeter and did not know the value of the resistor or the voltage of the battery. It would be quite a puzzling intellectual exercise! This is the greatest practical burden to overcome when learning magnetic circuits - we simply don't have the basic magnetic measurement tools like we have for electricity.
Ahhhh, but nobody can lay it out exactly like good old Karapetoff - whoever he was and where ever he now rests!