It works correctly either way -- externally with exit(Pid,kill) or by the process itself as exit(kill). The last one is just a shorthand for exit(self(), kill). Where self() is the process id of the currently running process.
But the way you showed was not one in which anyone was interested, synchronous exceptions work in more or less every language, and you can't assume readers know your self-killing is actually implemented via asynchronous exceptions since they don't know the language.
But the way you showed was not one in which anyone was interested, synchronous exceptions work in more or less every language, and you can't assume readers know your self-killing is actually implemented via asynchronous exceptions since they don't know the language.