Speaker cable arrows???

That's the way I have always read it. 

https://www.belden.com/blog/broadcast/understanding-skin-effect-and-frequency

So, you believe the “audio signal” travels inside the conductor, right? Regardless of whether the skin effect applies or not. I’ll ask again, the frequency of what?

It's moot where the "signal" moves, what matters is if the effects are relevant at the frequencies of interest in the system in which they are installed. All cables in a given system will have some directional aspects at high enough frequencies, but a 10 foot cable at 2/3 the speed of light means the signal takes 10 nanoseconds from the amp to the speakers, and neither the amplifier nor the speaker are a set impedance, and they certainly are not "matched".  Any transmission line effects will be 100's of db below the signal.

So what else could be "directional"?  Pretty much all that is left, absent intentional directionality (i.e. added RLC elements), is differences in lumped bulk RLC parameters based on the direction. However, the bulk RLC of any half-decent cable should be such that frequency dependent impact on transmission is a fraction of a db, meaning also no detectable phase shift by any human on the planet (we are terrible at detecting phase-shift that is the same on both channels), and since directional changes in the bulk RLC are orders of magnitude less that the bulk RLC, that fraction of a db now becomes thousandths of a db .... obviously undetectable.

It's fine to talk about single crystal copper (which improves by high temperature annealing, not cryogenic processing, but I digress), and it is recognized to have better conductivity at low frequency, there is no evidence of any frequency dependent effects that could impact audio transmission.

Also, the way I always see the skin effect stated, the “high frequencies” travel nearer to the surface - but not outside the conductor - whereas lower frequencies travel closer to the center of the conductor.

Is that completely wrong?

Not wrong, but not accurate either.

Electrical field penetrates conductor, moves charges, induces magnetic field, but the majority of the fields are outside the conductor, and the product of the fields, the power transfer is near 0 in the conductor.

At high frequencies, the electrical field does not penetrate very far into the conductor and hence the charge only near the surface of the conductor moves. The fields are still mainly outside the conductor and again, the product of the two fields (power) is near 0.

You said at high frequencies. Frequencies of what?

You also said audio transmission. What audio transmission are you referring to? The audio waveform? Current, voltage, Poynting vectors? Something else?

You also said the conductor material (and purity, I assume) makes a difference in the signal transmission. Why wouldn’t the non-symmetrical crystal structure of drawn wire also make a difference in the signal transmission? Especially if the non symmetry was visible on the surface of the wire. Yes? No?