When a horse's coat colour is subtly altered, but not as much as that the coat is considered a separate colour, we tend to use this a modifier. The genetics of two horses could be exactly the same, but when looking at them they seem completely different. There are a lot of factors that can cause the horses to look different, with not all of them having concluding research yet.
Grey is progressive, meaning a horse is born with little or no white hairs but gains them while he grows older. This process can be very fast, turning a colored horse into white within a few years, but it may also take a lifetime or even longer. This difference is at least partially hereditary, as the trait to lighten slow or fast tends to accumulate in breeds and/or lineages. But in every case, a horse will turn completely white eventually - the only exception is when it dies before reaching that stage.
When you speak of a horse with flaxen, you talk about a chestnut horse with a very light mane and tail. The shade of the colour differs from blonde to almost white. The colour of the mane and tail is always lighter than their body colour. While the coat remains within the normal chestnut range, it is possible the flaxen may more or less lighten the lower legs and belly of the horse, much like pangaré does.
It is not clear whether pangaré is caused by one gene or more, and whether it is completely recessive or dominant. Pangaré is often called mealy because of its mealy effects on the horse’s coat. Generally, it lightens the area above the muzzle, the flanks and belly, but it can lighten the entire coat as well. Perhaps it originated once as a camouflaging technique.
The mealy effect is seen on chestnut and bay based coats, but not on seal brown or black.
Sooty is most likely present in every horse breed and almost every horse shows the effects to some degree.
Black and bay based horses often have dark hairs and patches on the upper part of their body. This dark topline may look like a rather broad dorsal stripe, as is seen in dun horses. This effect is called countershading, as those parts of the horse are shaded which should actually lighten due to the sun. This might confuse predators.
However, sooty may also be responsible for dark liver chestnuts, which are often darkened completely or only from below, resulting in ‘muddy legs’. If it is, sooty shows a different overall pattern on chestnut based horses, than on any black based horses.
Sooty can also work on the mane and tail alone. There are no clear rules. The better rules you think of, the more exceptions you find. Sooty may be an overall term for multiple genes with multiple effects. Which is not unthinkable if you consider sooty may actually work as camouflage on a horse. I.e. protecting the horse from predators.