This is a bad definition. Self-awareness and "consciousness" is not a property that needs a definition like "soul" to define it, as it exists as a well-described mathematical concept. A lot of really important core work in mathematics has been done on the nature of "intelligence", "self-awareness", and similar concepts having to do with higher forms of sentience in the last ten years that have really nailed down these concepts in a rational mathematical framework (some rapid advances in machine intelligence have recently been made as a result, though those advances haven't leaked into the mainstream consciousness). Therefore, "soul" as you are using it is an intrinsic property of certain types of systems and does not need to be given a name or metaphysical connotations. It doesn't even exist as a separate construct in mathematics, but as a specific instantiation of other general mathematics.
I'm sure other people have different definitions of "soul" but this one can be discarded as mostly useless.