First off, it should be pointed out that the process of self-awareness you are describing is also experienced by human young. I believe we all get to experience that at some point in the age of 3 when we realize the influence of our thoughts over our behavior. Experiences such as "Mommy! I thought you went out!" can sooner or later guide the child towards introspective thinking.
Yet, I believe that relatively advanced level of realization is not the only experience that can be generally described as self-awareness. It can be reasonably argued, as does Rex Ker, it is hard to sharply distinguish between primal instinct and self-awareness as both phenomena generally suggest an "awareness" of external and even internal stimulus by the subject. A cat responds to the internal feeling of hunger and that suggests his is "aware" of its emotions but a cat doesn't seem to be ever able of critical introspection where a living thing contemplates upon its thoughts.
So considering that we humans do go beyond that primitive level of self-awareness, it is very intuitive that different levels of self-awareness emerge at different levels of the evolutionary process yet not all animals reach such advanced levels of self-consciousness as do human beings.
Now going back to the challenge of distinguishing different levels of self-awareness, it seems that the object of awareness is the real criteria for distinguishing different levels of self-awareness. Well, you might immediately respond that the object of awareness in self-awareness is evidently the "self", yet as in the cat's hunger example we learned that self-awareness is actually applied to an associate of the cat's self which was the feeling of hunger as experienced by the cat's self.
So I think the question rests fundamentally upon our full understanding of the nature of the conscious substance of living organisms including human beings. I, as an adherent to holistic theory of Mulla Sadra about the nature, origin, and development of human bodily and mental existence, believe that human self (or highest substance) just like every other living organism is a non-material being that interacts closely with his physical organs. Yet the multiplicity and compoundedness of his external physical existence introduces various kinds of perceptions to the non-material substance varying from primal instincts of hunger and lust, to external sensory perceptions (visual, audible, etc), mental abstractions and logical judgements thereupon which allows a wide and varied spectrum of further abstract conceptions and complex logical judgements.
Under this light, depending upon which particular set of perceptions (ranging from sensory and abstract perceptions to judgements and propositions) we predicate awareness upon, we get different levels/types of self-awareness as all types of perceptions become part of a living organism's self. Therefore, "I am angry" is as much indicative of self-awareness as is "There's a car out there", "I think Earth is round" and "2 + 2 makes 4" as they all indicate an awareness of a fact which is ultimately reflected in the subject's existence, as all of the prepositions can be preceded by "I am aware that..." So we can reasonably say that all awareness of conscious living organisms are also self-awareness.
But as an example as to how we proceed from the most elementary (primal) instincts to higher levels of consciousness let's trace different stages of the mental development of a new-born baby. When a baby first sucks upon her mother's breast, he is already aware of the existence of his lips (notice, that he still doesn't have a sense of the shape, proportion and other properties of his lips doesn't negate the fact he does sense their existence). And since he senses them as part of his existence, his self, that very primal instinct does qualify for a primitive level of self-awareness.
Taking the example further, the baby gradually establishes a relation between the perception of satisfaction (lack of hunger) after lactation and the perception of sucking on the mother's breast, and as a result craves for mother's hug once he is hungry again. That's how the child gradually arrives at such mental abstract categories (although not yet very articulate and vivid) as food (sensed as something that eliminates hunger, i.e. mother's breast), food-source (mother's breast or hug), eating (the act of sucking).
These vague more abstract concepts are reinforced upon the recurrence of similar experiences and later in the child's mental development come to be associated with visual images of the mother and her breast, and later with oral words such as "mommy", "breast", "food", "milk", "hug" and their respective referents and meanings, and as such lingual capabilities gradually evolve.
This introduces a higher level of self-awareness, that is, awareness of the most primitive abstract categorizations of immediately sensory percepts, all sensed by the subject's self. Reinforced senses of "me" and "others" on one hand, and reinforced concepts of existence and non-existence, cause, effect, etc on the other gradually enable the child to arrive at more philosophical questions or statements such as "Where did I come from?", "Why does apple fall?" etc. This is the stage where mental development can generate a myriad of new concepts, statements, questions etc and it is difficult to point out in particular what concepts and statements the child arrives at from this point forward, yet it can be safely said that since the child has already developed a strong sense of "me" and "other" enabling him to utter various first person sentences, the notion of self-awareness as recognition of the existence and impacts of thoughts is very likely already established.
This wisdom reveals that human self-awareness (which corresponds to his very mental substance) is constantly expanded as he acquires more knowledge of the world around him. Though, the knowledge absorbed, formed or arrived at can be divided into various types (i.e. primal, sensory, rational and spiritual) each with a particular impact on the subject's existence.
And in the meantime, it was also revealed that self-awareness is not passed from one living creature to the other, but it is a capability that separately evolves throughout mental development each individual conscious living being goes through.