To what extent can "The Black Cat" be considered a "horror" story?

Horror stories usually have some pretty common elements: 
Violence, the supernatural, dead things never quite dying, trustworthy characters
turning out to be untrustworthy, and murder.  "The Black Cat" has all of those elements,
to be sure.  We have a man who, at the beginning, is a kind, loving husband and
caregiver to animals.  By the end he is abusive and violent, even killing his wife
unexpectedly, and wreaking havoc on his pets.  He is violent, and the story has some
very gory details to it in describing the black cat's demise, and also his gruesome
murder of his wife.  There is the supernatural through the "visions" the man has of the
cat, and also in the end as the cat is apparently not dead, but alive and well in the
wall.  All of these factors lend themselves well to the horror
genre.

One last element of the horror genre that the story
has is a main character with a very interesting psychology.  At its heart, the story is
really a tale of a man who, through the abuse of alcohol, becomes victim to the demons
of his mind and his vices.  It is written through the narrator's eyes; he himself
describes his slow descent into madness and violence.  He can explain, quite lucidly at
a later date, what happened to him.  His mind starts playing tricks on him; he doubts
his sanity, and it leads him to do some pretty awful things.  When we are unsure of our
own mental soundness, everything is strange and bizarre, and we get to see that through
the eyes of the narrator himself.  His psychology is quite interesting, and despite his
descent into moral decay, he relates the events of the story quite clearly after the
fact.

I hope that helps; good luck!