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The first horse, Jenesaisquoi, is a liver chestnut because he is homozygous for sooty and heterozygous DP (dense pheomelanin). DP is not included in the gene test, so we have to infer its presence from the darkness of the horse compared to how many copies of sooty it has. Ammit has very kindly added liver chestnut to that result of the test, but anything thing else doesn't have a mention in the name (black chestnuts, black palominos, black buckskins which are all homozygous for both sooty and DP).

Bravado is a brown because he is homozygous for sooty and has the brown allele of agouti (At). At is the least dominant of the agouti alleles except for the recessive a; a horse that is Ata will always be brown, and can be distinguished by the almost black coloration on the neck. If a horse is AtAt it will be brown if it has any amount of sooty but bay if they only have the recessive sooty allele (S).

The darker bays will have at least one dominant Ssty allele and probably no pangere, which lightens the color behind the forelegs, in front of the hind legs, and on the muzzle. Your first bay is homozygous for sooty and hasn't been rerolled since Ammit reworked bays in the new image generator. The second one definitely has one dominant pangere and one sooty and so falls into the mid-range of bay variations with the pangere lightening. The third one has no sooty and no pangere. The lighter legs on the last one is one of the random variations on the bay coloration. A bay with even lighter legs than that will probably have the fourth agouti allele, A+ or wild bay.

http://www.huntandjump.com/horse.php?horseid=3104751
Wild bay is dominant over all other agouti alleles.