This Horse Bullshit (Horseshit?) Explained, Briefly and Incompletely:
Horse coat genetics rely on a handful of coat genes but lots of genes modify those coat genes. The first layer is pretty straightforward. Horses have two genes that produce one of three base coat colors: Chestnut (reddish brown), bay (brown), and black (...black). The intensity (shade) of these colors vary, but it's fundamentally just those three.
Then there are a ton of genes that dilute those colors. Some of those genes are dominant, others are recessive, and others are dosage-dependent (one copy will have an effect, but two copies produce a stronger effect). There are also other genes that produce white within hairs or produce specific distribution patterns.
The specific coat + modifier combinations have specific outcomes, so horse people have gotten very granular with their descriptive terms and it looks utterly incomprehensible to an outsider. Generally, the muzzle and eyes will tell you a lot, but sometimes it truly does come down to needing a DNA test or pictures of the parents.
Flaxen liver chestnut vs. silver dapple: Liver is a shade of chestnut, and flaxen horses are horses that have a particular gene that causes lighter manes/tails than coats. However there are other genes can also cause lighter manes/tails too—silver dapple is one of them. Silver dapple dilutes a normally bay or black horse so that you have a flaxen effect on the mane/tail and sometimes also a dappled (spotted) coat. These are known to be stupid hard to distinguish in some cases but genetically, a FLC has a chestnut base color and silver dapple has black or bay.
Mealy vs. pangaré: ...ngl I have no idea what the difference is here (op help??). Regardless, mealy/pangaré are patterns that produce lighter hairs around the eyes, nose, belly, and legs. It's a common gene and can show up in addition to most any other coat pattern.
Flaxen chestnut vs. dark palomino: Flaxen chestnut is same as above, but with a lighter (non-liver) shade of chestnut. Palominos are also chestnut horses but carry a single cream gene, which dilutes them to give them that lighter mane/tail. (If a chestnut horse has two cream copies, it's cremello. A bay horse with one cream copy is buckskin, and two copies is perlino.)
Perlino vs. amber champagne: Perlinos have pink skin without much freckling, so their coat looks really consistent—and they also have blue eyes, thanks to that double cream copy. Champagnes have pink skin with dark freckles—note the pink nose/lips and darker spotting on the ribs and thigh. They also usually have brown eyes by the time they're adults. (To my knowledge, the champagne and perlino effects are also notable within specific breeds—so if you were hypothetically very knowledgeable, you might know whether the genes were champagne or perlino without really needing to investigate.)
Dun vs. buckskin: Notice the stripe on the spine of the dun horse. It's a little hard to distinguish on the photo, but duns also have bars (stripes) on their legs—you can see a couple darker stripes near where the black leg blends into the main coat. If you /don't/ see the dorsal stripe or bars, it's a buckskin...probably. Other genes can sometimes also make or hide those effects. DNA tests are helpful here.
Hopefully that's mostly correct/intelligible; it's a little simplified. Horse genetics are one of those things where it initially seems super complex, then you learn a bit and it starts to seem controllable...so you read a little more and it's back to being too complex.
