I read the 'starting Point' link you posted, the black horse description seems to fit Breeze, before she sweat a lot on the first really hot day this spring, she was black all over, not one brown spot.
I would just test for agouti unless you want to know whether she's homozygous EE or heterozygous Ee on the extension/red allele.
The cheapest route would be the 'regular' agouti test either from Animal Genetics which you linked to or through UC Davis (that's where I tested my horse). Either one is $25. Their tests don't distinguish between classic bay (A) and seal bay (At) but you can make an educated guess from her visual appearance that if she's not black (aa) then any A's that show up are actually At's.
The most "complete" test would be PetDNA's "Comprehensive Agouti = ‘a’ plus ‘At’ combined" but it'll cost you $60.
I just sent in the sample! I don't care if she is a true black or not, I will still love her.
I do however have a question. Aa means she is a true black, but what does Aa and AA mean? Does Aa mean she has a black factor, but is a bay, and AA means she is just a brown horse? This is what is confusing to me!