Monster can have two definitions. It is possible to meet one definition without meeting the other, and still be considered a monster. You might have a monstrous appearance, yet not commit monstrous acts. Vice versa is possible, too. You don’t need to meet both definitions to be considered a monster.
You seem to be more focused on actions. I would argue that Rek’sai is a monster, regardless of her level of self-awareness. She’s part of an invading force that wants to end Runeterra. Whether she’s aware of her own purpose or not, that makes her more monstrous than another creature - such as a tiger - that doesn’t know any better.
When labelling someone a monster because of their actions, that definition depends on the observer’s moral code. Few people self-identify as a monster; it’s a label given by other people. We judge what’s considered monstrous based off what actions we think are extremely unacceptable. Mental illness doesn’t always exempt someone from being considered a monster - in fact, sometimes it adds to the idea. Jhin, therefore, would meet our understanding of monster.