I agree with Mown in that horror is supposed to leave you with chills, not to just freak you out or make you jump. Death in horror should be the moment of catharsis -- the point where all the tension that's been built up is released -- but it is not necessary by any means.
Note the following examples are from media I have not directly experienced.
H.P. Lovecraft's work, especially on the Cthulhu mythos, focused a lot on the creeping fear of both the unknown and of the processes and events that are much bigger than any person, with the ultimate goal usually to reduce the reader to feeling so very scared of the much grander universe beyond their knowledge, mostly without death being a focus. There are, after all, fates much worse than death.
I am also reminded of a quote from, apparently, Stephen King, "Nothing is so frightening as what's behind a closed door." Just having a death or something disgusting "on screen" as it were does not induce horror because it does nothing to ratchet up the tension on its own.
horror is a word that has an extremely malleable meaning when used in real life to refer to fiction. anything with dark elements will generally be called horror by a large number of people.
I'm always of two minds whenever semantics are brought up.
One side of me is saying, in Bob Chipman's voice, "WORDS MEAN THINGS".
Another side of me is saying, "yes, language is a thing, that's why literally means figuratively
in the dictionary".
I think a large part of the problem is that mass media puts labels like these in the way
they want people to think, and as people are exposed to more and more of those examples (especially younger audiences who are more susceptible to such suggestion), then it becomes the common usage of the term because it's what we've been conditioned to expect. The thing is this doesn't create a meaningful distinction to those wanting to discuss it (and also has the added frustration of needing to supplement the original meaning in order to differentiate it from the new one). In times like these I usually default to referring to those who would be deemed suitable references: in the case of defining horror, authors held up as horror writers; in the case of defining a furry character (a discussion I recently had in the meat), people actually in the furry fandom; and so on. I hate either relying on dictionary definitions or going with the "3 million people can't be wrong" mentality.