They'd be classified as crimes of passion.
"A crime of passion, or crime passionnel, in popular usage, refers to a violent crime, especially murder, in which the perpetrator commits the act against someone because of sudden strong impulse such as sudden rage or heartbreak rather than as a premeditated crime."
Again, the difference is his intent. By going for this person's throat, he was attempting to murder them.
This is disregarding the fact that he's proven himself a blatant threat to the school as a whole already, in what's meant to be civilized lands. A fight on the first day, almost killing people with uncontrolled telekinesis on the next, and intending to kill another a few short minutes later. At the very, very least, I'd give him a device that inhibits his powers and have a powerful Youkai constantly at his side.