Champagnes have weird color names.
Gold Champagne = Chestnut (ee ??) + Champagne gene
Amber Champagne = Bay (E?A?) + Champagne gene
Sable Champagne = Brown (E?At?) + Champagne gene
Classic Champagne = Black (E?aa)+ Champagne gene
So the "Amber" part of her color just means she would be a plain bay horse without the Champagne gene. She's E(black-based) A(agouti) plus Ch(Champagne). She also has that At gene in there which doesn't show on her, but means she could have a brown foal.
The Extension (E/e) gene just determines if a horse is black-based or red-based. If the horse has at least one E gene it's black-based, if ee it's red-based. All horse colors are one or the other with modifiers added for colors besides Black and Chestnut.
The Agouti (A/At/a) gene here is considered only active in black-based horses, so a horse that is red-based (ee) won't express it no matter what alleles are present. What it does is restrict where on the horse black pigment is fully expressed. It limits actual black to the "points" of the horse and leaves the body lacking enough pigment to be truly black, so it's just sort of a faded black color (which appears red, but isn't). So a horse with E?A? is the full-on Agouti color we call "bay" with black points and reddish body. A horse with E?At? has more expression of pigment on the body, has brown "highlights" on the nose, flank, etc and black points and is called "brown". A horse with E?aa is black all over with no restriction from the Agouti gene.
So now we have what E/e and A/At/a are and what they mean to horse colors, you can take all those and apply the modifiers like the white-spotting patterns (Tobiano, Sabino, Frame), Roan, Dun, etc and add in that Champagne has those weird names and so does Cream(CcrC) (Chestnut + Cream = Palomino, Bay + Cream = Buckskin, Brown + Cream = Brown Buckskin, Black + Cream = Smoky Black) and Double Cream(CcrCcr) (Chestnut = Cremello, Bay/Brown = Perlino, Black = Smoky Cream).
Most of the genes here are dominantly-expressed (the capital letter means the horse has that trait), but Pearl is recessive (a horse must have prlprl to be Pearl) and Agouti has 3 alleles, where A overrides the other two, At only overrides a and aa means "non-Agouti).
Hope that helps some people who are still a bit confused. Variable color names don't help. It's like the gene that's called merle in every other dog breed is called dapple in dachshunds. You just have to learn the color names and roll with it.