Gore certainly includes the bloody violence in zombie movies, but it can also refer to real-life violence, especially bloodshed and murder. If you stabbed someone with a sword, you gored them. An animal with sharp horns, like a bull, can gore a person to death. Gore is also blood that's clotting in a wound.
Hemophobia refers to the intense and irrational fear of blood that interferes with an individual's ability to function in their day-to-day lives for at least 6 months. Currently, it is listed under specific phobias in the Diagnostic and Statistical Manual of Mental Disorders (DSM-V) as blood-injection-injury phobias.