I made another comment pointing this out for a similar definition, but OK so awareness is being able to "recognize", and recognize in turn means "To realize or discover the nature of something" (using Wiktionary, but pick your favorite dictionary), and "realize" means "To become aware of or understand", completing the loop. I point that out, because IMO the circularity means the whole thing is useless from an empirical perspective and should be discarded. I also think qualia is just philosophical navel-gazing for what it's worth, much like common definitions of "awareness". I think it's perfectly possible in theory to read someone's brain to see how something is represented and then twiddle someone else's brain in the same way to cause the same experience, or compare the two to see if they're equivalent.
As far as a computer process recognizing itself, it certainly can compare itself to other processes. It can e.g. iterate through the list of processes and kill everything that isn't itself. It can look at processes and say "this other process consumes more memory than I do". It's super primitive and hardcoded, but why doesn't that count? I also thinking learning is separate but related. If we take the definition of "consciousness" as a world model or representation, learning is simply how you expand that world model based on input. Something can have a world model without any ability to learn, such as a chess engine. It models chess very well and better than humans, but is incapable of learning anything else, i.e. expanding its world model beyond chess.
If you created a computer program capable of learning patterns in the behavior of its own process(es) and learning how those behaviors are similar/dissimilar or connected to those of other processes, then yes, I’d say your program is capable of consciousness. But just adding the ability to detect its process id is simply like adding another built in sense; it doesn’t create conscious self awareness.
I think we largely agree then, other than my quibble about learning not being necessary. A lot of people want to reject the idea of machines being conscious, but I've reached the "Sure, why not?" stage. To be a useful definition though, we need to go beyond that and start asking questions like "Conscious of what?"
I think the definition of consciousness meaning "internal state that observably correlates to external state" would clarify here. Gravel wouldn't be conscious, because it has no internal state that we can point to and say it correlates to external state. Galaxies/the universe doesn't either, as far as we can tell. Galaxies don't have internal state that represents e.g. other galaxies, other than including humans in that definition, but it would be more proper IMO to limit the definition the minimum amount of state possible. You don't count the galaxy as having internal state that represents external state, if you can limit that definition to one tiny, self-contained part of the galaxy, i.e. a human brain.