I've been thinking that maybe it would be a good thing to get into some of the details of my thought process and why the materials and construction matter to me so much in this case. This might be rambling a bit, so please bear with me. And please remember, that a lot of what I'm talking about here isn't related to PAFs specifically but to all pickups in general. Lastly, I'm trying to keep it simple, so I may be saying some things that are pretty obvious.
All of this is just my opinion, so if there's anything in this that isn't correct, please let me know. There is always more to learn, and I don't claim to be an expert here.
Pickups are simple devices that transduce the vibration caused by the strings into an electric signal which can then be amplified. That's fairly obvious, but worth pointing out at the beginning. These devices contain only a few parts that are necessary and the structures that are physically required to support them.
The magnetic field is the key component, because without this, the only way that the coils could sense the motion of the strings is through physical vibration. It would be too weak and very prone to feedback. So, the magnetic field is king and anything/everything that effects it is going to affect the sound.
The source of the magnetic field is the magnet and many different magnets have been used (including an electromagnet that required a voltage source). It's fairly well accepted that different types of magnets produce different sounds (e.g. ceramic, alnico, cunife, neo) because of how they project a magnetic field. There are different grades of each (e.g. A2, A3, A4, C8, etc.). Each of these grades is separated by an average value and a range of tolerance. Say 0-9 is "A", 10-19 is "B", 20-29 is "C" and so on (magnet grades use numbers, which is where we get A2, A3, etc.).
Considering the effect on the size, shape, and strength of the magnetic field, the size and shape of the magnet are also important and do affect the sound. Multiply that for designs that use multiple magnets such as P90s, Dirty Fingers, most Fenders.
Some magnets may produce small, strong fields that decay quickly, others may produce broad, diffuse fields that decay gradually. The details aren't necessarily important unless you're a physicist, but it is important to keep the size, shape, and density of the field in mind because this will have an effect on the sound and is why different magnets sound different.
Tolerances within each grade can significantly change the final result, too. Using my "A, B, C" example above, a 19 would be a "B" and is much further away from a 10, also a "B," than it is to a 20, which would be a "C" but only slightly different from the 19 "B." So, the borders matter because while they help us classify which is which, they also aren't as descriptive as we tend to think they are. This is multiplied exponentially by the ratios of the metals in a given alloy. If the average is, say 15, and you've got 10 (in spec) of aluminum, 19 (in spec) of nickel, and 15 (average) of cobalt, it's not really the same as 15 AL, 19 NI, and 10 CO. I'm making the numbers up in an attempt to demonstrate how variable amounts can remain "in spec" but still change the resulting alloy. I'm not a chemist or metallurgist, but I think this is still a valid point.
Another thing to consider here is manufacturing tolerances, which can be anything agreed to between the manufacturer of the magnet and the consumer of that magnet (pickup company, reseller, importer, any number of middle agents between where the magnet is made and the company that uses it to make pickups). These tolerances, say 10% can take an example of 10-10-10 (again using my "A, B, C" example above) and turn it into a 9-9-9 which results in something being labeled a "B" but is actually an "A."
So then there's the coil. The magnet magnetizes the string and the string moves, interacting with the stationary magnet to induce current in the wire, which is wound in a coil to provide more wire to produce more current. Since the magnetic field varies in strength as it moves away from the source magnet, the size, shape, and density of the coil is going to greatly affect how well the vibrating string transduces its mechanical energy into electrical current. It's fairly well accepted that different gauges of wire will produce different results, but...
What about the shape of the coil? Jazzmaster pickups don't sound like Stratocaster pickups in part due to the shape of the coil (wide and thin vs narrow and thick). Jazzmasters don't sound like P90s, either, in part due to the different coil shapes (both wide, but one thicker than the other).
Then, there's how the coil is wound. Is it very tightly wound or is it more loosely wound? Too loose and it will allow for too much movement from mechanical sources other than the string that will also induce current. This is where microphonic feedback comes from. Too tight and it will restrict that movement, which can also include the guitar itself vibrating. This is why some pickups allow more of the acoustic sound of the guitar to come through while others sound the same (or very similar) no matter what guitar they are installed in. Potting will effectively stop almost all movement in the coil itself, which will help eliminate microphonic feedback, but also kills any chance of the pickup sensing the acoustic qualities of the guitar itself.
Is the coil symmetrical or is it asymmetrical? Some coils are wound like hourglasses, others may be wound heavier on the top or on the bottom. Some may be heaviest in the middle, more like a rounded ball shape. Equally important but not as apparent is what shape is the inside of the coil? Just because the inside of the bobbin is flat (if it is), doesn't mean that the wire fits snuggly against it. A good example of this is the lead wires attached to the beginning of some coils and then wound part-way around the coil before the magnet wire is wound. Depending on the size and insulation of this lead wire, the effect on the inner shape of the coil can be significant. What about a figure-eight shaped coil? If you've got two mid-focused coils stacked on top of each other, as in some hum-cancelling "single-coils," could that coil shape contribute to why they don't ever quite sound like true single coils? Maybe...
What about the magnet wire itself? Is it evenly within its gauge or does it vary, as specification and manufacturing tolerances apply here, too. What is the insulation made of (e.g. plain enamel, polyester, etc.), how thickly, and how evenly is it applied? All of this will affect the shape, size, and density of the coil.
Scatter-winding, I think, is just another way of saying the layers of the coil are more or less symmetrical. All mechanical devices, like pickup winders, will exhibit a greater or lesser degree of accuracy. CNC winders can be very accurate, but can be programmed to include specific deviations from a completely symmetrical layer. Hand-winding is the least consistent. Mechanical winders are somewhere in the middle. They all work a little differently, and all ultimately help determine the shape, size, and density of the coil.
What about the bobbin itself? Most are simple oblong cylinders. Age may cause them to partially collapse, which will change the shape and the internal tension of the coil. Does what the bobbin is made out of matter? Well, in general, that material, some type of plastic generally, isn't magnetic. So, unless it is or somehow influences the magnetic field (like changing shape over time or resistance to damage), probably not. The shape of the bobbin is what matters, including any deformations.
So, then we get into the supporting structures, like the baseplate, covers, the pole slugs, and/or the pole screws. The poles, slugs or screws, have to be made of magnetically conductive material or they can't serve their purpose of conducting the magnetic field up to the strings. Sometimes, they are the magnets themselves, as in most Fender pickups. Sometimes, they are made of some variety of steel, as in most Gibson-style pickups. In any case, the composition of that material is bound to have an effect on the resulting magnetic field.
The baseplates, while below the magnet, coils, and strings, also have an influence on the magnetic field. If they're made from magnetically conductive material, the composition of that material will also affect their influence on the magnetic field. If they're made from something that is not magnetically conductive, then the magnetic field is still influenced as it is no longer being partially reflected back up into the coil and the strings. Think of Telecaster pickups that have a metal "reflector" plate on the bottom or adding one of these reflecting plates to a Strat pickup, which is a common option from many winders. Does that change the sound? Yep.
Same thing for covers. During the late '60s and '70s, when Les Pauls and humbuckers were gaining popularity, many players, like Jimmy Page, removed one of both of the covers from their humbuckers. Supposedly, this increased treble in a time when treble helped cut through crowd noise and insufficient PAs. I'd think that the presence or absence of a magnetic material between the coils and the strings will absolutely contribute to the resulting sound and the composition of that material plays a part in that result, too.
To be continued...