I use the word "self-consciousness" for that. For me, consciousness means awareness of the environment without any implied awareness of the self. I lizard is conscious. Is it aware of its own existence? Maybe. How about a fish? They are probably not aware of themselves as such. If not, they are still conscious - except when sleeping. As I use the words, consciousness is more like sentience, vigilance or wakefulness than self-awareness.Consciousness is being aware of one's existence ..
I'm not sure that Parnia is the guilty one here. He may be doing good science and using language appropriately. It may be only zealous Christians that are claiming that he is looking for the Christian soul, or that he claims to have found it... by juxtaposing the words, slanting his writing, Dr. Parnia is using slight of hand to convince the reader that he has proven a soul. For me, that's not science at its best ..
I read several links by and about him, plus looked at a YouTube about him, and never saw him overstep the bounds of professional propriety, although he may well have done so elsewhere.