Which one is used more often? You can cut your leg, hand etc. but do you slit or cut your wrists? Is the correct word related to the method of action that causes wrists to bleed?
You can cut, slit, or slash your wrists, and the denotations are all the same. A Google Ngram shows that the first two are currently equally popular and the third is only a little less popular.
Each word has different connotations for a careful and caring writer.
An old girlfriend of mine cut her wrists and was hauled off to the U of Iowa’s mental hospital about 40 years ago. She used a dull razor blade, made very shallow cuts, and failed to end her life.
Potential suicides who really do want to die usually slit their wrists with a sharp knife or razor blade, and they usually succeed in opening a couple of veins and maybe an artery.
Battered women are often described as having slashed their wrists, which gives the impression that they used a machete, as primitive slash-and-burn farmers did and still do to fell big bushes and small trees.
The level of self-inflicted violence is much higher in the word slash than in the words slit and cut.