Horror as in horror. I don't know. Perhaps a "thriller" would be a closer term to what I'm thinking (remember, English as a second language here). We have tags covering most facets of emotion a story can elicit in response - happiness, sadness, repulsion and hate (in many many forms), and this one would concern fear, on either side of the fourth wall. I don't know if it's even possible to make the reader fear in sympathetic response to a written story...
That is exactly the problem. I don't think it's possible, really.
Also, there are many facets of fear and most of them have already been covered by our large collection of incredibly horrible tags. Although you can, of course, have fear without having any violence, gore or death. In fact, that's the
best kind of horror. But that doesn't change the fact that the horror tag would be added to most of the stories so far where we have characters fearing for their lives.
I think I'll add it regardless, since horror without gore or death is most definitely possible.
Also, Flying Dice, you are a freaking ninja. Of course there are differences in horror. That's why we have lower- and upper-case, after all. If the characters are genuinely afraid, even if it is just a Scooby Doo sort of story, I would probably add the horror tag. Upper-case horror tag would be much more serious. Also, just because nothing physically bad happens to the main character(s) doesn't mean that it can't be incredibly fucking scary.
Now that I think about it, "horror" has too many connotations and different interpretations. "Fear" would probably be more fitting.