I think the basis of "self"-awareness may be the understanding of oneself in the abstract, as a part of the abstractions that one categorizes qualia with. {A rambling aside: by "qualia" I understand the non-verbal information within consciousness: sense qualia: sight, sound, touch, taste, smell; and embodied 'gut' (prioperceptive) qualia, informing consciousness of one's immediate, emotional reaction. Qualia are a 'language': part of the internal signalling between competing cognitive systems. In computational terms: the current environment is conscious input to unconscious sense algorithms which output sense qualia; which are input to an unconscious gut algorithm, which outputs to consciousness a gut-reaction emotional qualia (the feeling of fear, e.g., if your current environment includes a ticking timebomb); the gut reaction qualia trigger memory algorithms which output a more informed rational appraisal of the current environment ("I know I saw MacGyver dismantle one of these once"); the emotional reaction and rational appraisal are input to behavior algorithms ("10 seconds... do I trust MacGyver and cut the blue wire, or run like heck?"); cue the anxiety qualia, cue the faith in MacGyver qualia, behavior algorithm scales tip in latter's favor ("cut the blue wire"), decision qualia cues doubt algorithm ("no wait, the red wire, the red wire, dummy!"), memory algorithm ("yes, it was the red wire!"), sense algorithm ("2 seconds!"), gut algorithm ("this is it, chump!"), and... tune in next week.
Note that the output described in words isn't necessarily output to consciousness in words: the output is probably better characterized as a mix of feelings of anticipation -- potential success or disaster -- that involve any or all of the sense and gut qualia, which we might call irrational qualia; and possible courses of action, understood as urges to move here with this instrument in hand and do this, in equivalent terms that is, not those exact words.
Words of course allow us to step back from our environment and consider it in abstract terms, try to classify it and compare it to previously learned, formally similar environments. If the information in memory isn't sufficient, we can inform consciousness with information stored outside memory, peers, books etal., which abstract others' understood experience as potential knowledge; then commit that understood experience by rote, or abstract the logic behind the experience and apply it to other formally similar situations, etc.}
Anyway, the key point I want to make about "self-awareness" is that
we are self-aware when we understand ourselves as an abstraction -- "one locus of experience, a point-of-view among other points-of-view, agent among agents, differentiating self from not-self and other selves, embodied within time & space, etc." -- and are able to see ourselves in abstractly understood potential future situations, to make rational plans based on these abstractions which include our "self" as potentially effective and/or vulnerable agent, and thus behave rationally, taking account of our understood past and potential futures, rather than simply react instinctively, or by rote ritual, to our sensed present.
Hope that makes sense and isn't too obvious or irrelevant.