It's a fine definition for an abstract. If it contained a full, precise definition, it wouldn't be an abstract. It's a single paragraph that summarizes a dense, academic 30 page paper. Everyone who reads them understands that the abstract is about the general idea, not formal correctness.
It is not. An abstract is one of the most important parts in a paper and that's why authors tend to avoid writing definitions on them but stating something like "A definition for X is given in the paper". It is meant to "describe" the paper rather than a summary or introduction of the subject in question.
In any case, writing a good abstract is hard, and it doesn't diminish the quality of this paper.
