Let me get technical with you
A black horse will have the following genotype: E_ aa crcr
This means that the horse has at least one copy of the black extension - E. Two EE means they are homozygous for black. ee would be a red based horse, basically a chestnut.
To be black and not bay or brown, the horse needs to be aa. This means they have no copy of the agouti gene. There are three definite types of agouti gene, and two of them have been isolated - bay (A) and brown (At). The other is wild bay (A+) and hasn't been isolated yet. The current theory leans toward a simple dominance of agouti - A+ A At. So to be wild bay, the horse only needs one copy of the A+ gene, to be "plain" bay they need an A gene and not to have an A+ gene, and to be brown they need an At gene and no other agouti gene (I am not sure this is the case, but that's personal opinion so I will try to leave it be lol). Any copy of the agouti gene means the horse is no longer "black" and is some type of bay.
Then we have the cream gene. crcr is no copies of cream, and the horse will be the colour designated by it's other genes, in this case, black. Give them a single cream, and they become Crcr, and a single dilute. This can be called lots of things - smoky black and black buckskin are the two most common, and most people that know the colour will understand if you use either of those names. If the horse has two copies of the cream gene, they are CrCr, and are a double dilute. Most people call a double dilute black a Smoky Cream, but I have heard it being referred to as a black perlino.