How does magnetic hysteresis (ie. the B-H curve) transform the input-output of a guitar pickup or speaker?

I understand that magnetically dependent equipment like a guitar pickup or speaker are affected by magnetic hysteresis.

Here is an article discussing the effect on speakers for example. That article summarizes its etiology pretty well and shows a B-H curve they measured of:

As you can see there is a gap between the upswing and downswing (hysteresis) as well as a very strong saturation effect in flux density (B) when driven to major extremes of field strength (H).

A guitar pickup should also be subject to the same hysteresis and saturation effects of magnetism. A pickup’s function is summarized as:

A typical magnetic pickup is a transducer (specifically a variable reluctance sensor) that consists of one or more permanent magnets (usually alnico or ferrite) wrapped with a coil of several thousand turns of fine enameled copper wire. The magnet creates a magnetic field which is focused by the pickup’s pole piece or pieces.1 The permanent magnet in the pickup magnetises the guitar string above it. So the string is, in essence, a magnet itself and its magnetic field is in alignment with that of the permanent magnet that magnetized it. When the string is plucked, the magnetic field around it moves up and down with the string. This moving magnetic field induces a current in the coil of the pickup as described by Faraday’s law of induction.2 Typical output might be 100–300 millivolts.

So I believe a guitar pickup is measuring fluctuations in the magnetic field as the string moves up and down within the range of the coil. Faraday’s law states that the electromotive force (EMF) is given by the rate of change of the magnetic flux:

$E = − frac{d Φ}{d t}$

where $E$ is the electromotive force (EMF) and Φ is the magnetic flux. I believe the EMF represents directly the voltage out of a guitar pickup for example.

Let’s say you are trying to simulate the input-output transform from the BH hysteresis/saturation curve of an audio signal going through a magnetic device like a speaker or guitar pickup. For a speaker you would have as input the audio signal and for a pickup the amplitude/velocity of motion of the string fragment moving up and down within the range of the transducer or magnetic field.

How do these "input amplitudes" relate to the BH curves to dictate the output effect?

What I mean is for example, if the $x$ axis of a BH curve relates to your input signal, and the $y$ axis relates to the output, what is the relationship? Could $x$ (H) be taken directly as your input signal (input audio for speaker or amplitude/position of string fragment for pickup) and $y$ as your output signal directly?

The part that throws me off is they say EMF is the time derivative of magnetic flux. If for the pickup, H (magnetic field strength) is directly related to the amplitude of the string position, then you transform that amplitude of string motion by the BH curve into B (flux density), you’d then have to take the time derivative of B to get EMF, which will no longer be an "audio signal."

The only way I see to get an audio signal out is to transform the input/output directly (ie. string position in (H) = voltage out (B)).

How does this work? Thanks.