I guess there are many ways to cut the carrot. What matters is the liberating understanding, the liberating glimpse.
The inquiry into reality, into the true nature of I, the true nature of the world body mind, the inquiry which brings liberation from the illusion of separation which plagues us is what matters.
In my lingo, I refer to mind states as manifestation, as appearances. We do make a ‘conceptual’ distinction between waking, dream and deep sleep states, but such distinctions are pedagogical and do not exist in reality. We do make a distinction between outward directed and inward directed consciousness, but once again, such distinction have no reality.
The liberating glimpse is in the understanding that there is one reality. The reality of consciousness/awareness, known by many names such as Brahman, the Absolute, God, Being, etc.
I refer to the body mind as endowed with sentience (rather than consciousness).
There are various degrees of sentience. Under anesthesia, the body mind’s sentience differs from the sentience when it is not under anesthesia. Irrespective of the sentience of the body, consciousness/awareness/reality remains unchanged, unaffected by the phenomenal realm. In the common lingo, people refer to an anesthetized body as ‘not conscious’. I refer to an anesthetized body as having a different form of sentience. The senses are dimmed down under anesthesia in order not to perceive the surgery.
It seems that Nisargadatta makes a similar distinction between the terms awareness and consciousness, similar to yours. In my experience, this has led to much confusion given in many languages, consciousness and awareness go by the same word.
Having said all that, the language which serves someone to arrive at the liberating glimpse is the correct language. Whichever form this language takes is of no importance except for pundits, which we are not.
I am familiar with this model. In my sharing, I refer to the term awareness as an attribute of consciousness. Awareness as the knowing aspect of being. Wherever there is being, there is awareness and wherever there is awareness, there is being.
In the Indian tradition, they refer to Sat/Chit/Ananda as the summation of the non-dual understanding. Awareness/knowingness is Chit and consciousness/reality/being is Sat.
Sat/Chit/Ananda invites the understanding that awareness/knowingness and consciousness/beingness are not two.
Again, all of this is pedagogical. The formulation that leads you to the liberating glimpse is the correct formulation.