Is the person aware (not verbally, but on the gut level) that their own skills could improve in the future, or do they implicitly assume that their skills will always stay the same?

I think that's a good definition of the theory as Carol Dweck would define it; I'm just not so sure that's the best definition of the experimental results. For instance, what precisely is gut level awareness? How would I test it experimentally if they can't vocally express this awareness? Is the fixed mindset due to unawareness of the ability to improve or due to a desire to stay the same? Is it that the individual is aware they can improve, but simply is overestimating their own probability of getting worse or underestimating their probability of getting better? Is it an issue of avoidance of failure or is it a failure to approach goals? If I was to define the two terms, I might use something like:

fixed-mindset - When individuals are praised for their attributes, they are more likely to engage in behaviors intended to display or protect those attributes. growth-mindset - When individuals are praised for their effort, they are more likely to engage in behaviors intended to improve their attributes.

But that's rough. I'm not familiar with all the studies on the subject.