You can call it anything you want....no problem.
But consciousness is seen normally as some external quality ....in an objective sense. The Self is just .....the subjective self.....each of us. It is a better way of identifying with it rather than see it as some external entity. There is nothing religious about it...!
Essentially IMO...consciousness is an attribute or property of the Self.
Firstly, there is plenty of evidence that there is a state called consciousness(i.e. the mind's awareness of the world, which may or may not include awareness of itself)
Secondly, all the evidence suggests that the state of consciousness is limited to the neural processes within the organism which produces it. Hence it is a subjective state, not an objective state.
Thirdly, the sense of self is just the way that a human being describes his/her awareness of the state of himself/herself.
Fourthly, there is no evidence that Self is an actual entity, whether that be in either a subjective or objective setting. Rather than consciousness being an attribute or property of something distinct which you seem to be calling 'Self', it is much more likely that the sense of 'Self' is a product of the self awareness aspect of consciousness which human beings display.
If the brain is impaired, then there is a risk of the mind being impaired, and, as it has been shown in numerous instances, this can lead to an alteration or, in the case of traumatic injury, even the disappearance of this sense of 'Self' completely with the accompanying loss of awareness. Of course, correlation does not necessarily mean causation, but, unless this 'formless essence' of Self, (as Ekim describes it) has hard evidence to support it, then, like the smoking correlation with lung cancer, I'll go with the correlation until that day arrives.