It is certainly not a full definition; one might reasonably expect a "real" programming language to have not merely an abstract mathematical model defining the language's semantics, but also a concrete implementation of a compiler or an interpreter which runs on an actual computer. Then again, one might reasonably not expect this. Is lambda calculus, for example, a programming language? Wikipedia doesn't define it as one, but if somebody else does then I can't say they are wrong.
There are many aspects of what makes a language a "programming language"; reasonable people might disagree on whether a certain aspect (say, Turing completeness) is strictly necessary for something to count as a true programming language. It is unlikely that any list of such aspects would ever be agreed by all computer scientists, or even just the academic computer science community, to be both correct and complete.
The concept of a "programming language" is not in the same category as mathematical concepts such as "group". It is possible for a formal definition of a group to be correct or incorrect, but a formal definition of a programming language can merely be useful or not useful (or, more useful or less useful) relative to the use at hand. Presumably, the definition you quoted was useful for the author of the paper you found it in.