Correct- there are three known agouti mutations. Classic bay, wild bay, and seal bay (aka seal brown, brown, dark bay, black bay, etc). All of these affect black based horses by restricting black on the coat. All of these will have black manes and tails and black (to a varying degree) on the legs, unless some other gene is at play (pinto patterns, silver gene, etc.)
Wild bay restricts black the most- the mane and tail are black, but the black on the legs stops very low, just around the fetlock. The rest of the body is a uniform reddish color (the exact shade can vary from horse to horse, but will be fairly consistent from season to season on the same horse) Wild bay is uncommon and is thought to be recessive to classic and seal bay. Classic bay is very similar in characteristics to wild bay (body is a uniform color without a lot of seasonal variation), but with the black going up higher on the legs, generally over the knees/hocks.
Seal bay restricts black the least. It's trademark features are a high degree of seasonal variation and the lighter "soft points," especially on the winter coat. Seal bay horses are often so light in the summer that the appear to be classic bay, but darken in the winter. Some are nearly black (except for those tell-tale soft points!) year round.