Is there a formal definition of signature of natural selection?
Here is how I would personally explain what a genetic signature is:



A genetic signature of a certain type of evolutionary history (type of selection or demography) refers to the effect that this specific type of evolutionary history has on genomic properties of populations today.

We are interested in understanding such signatures because knowing the association between a number of evolutionary histories and their genetic signature allows us to reconstruct the evolutionary history of a population just by looking at the genomics of this population.

For example, recent selective sweep, population expansion after a recent bottleneck and/or linkage to a swept gene are all causing negative Tajima's D. Balancing selection and/or sudden population contraction are causing positive Tajima's D. In this example Tajima's D is a statistics that is used to measure this so-called signature of the different process.



The above was NOT a citation. I am NOT aware of any publications defining rigorously what a genetic signature is. I think it is not so uncommon for authors to use a term without strictly defining it. As long as the unclarity of the definition does not yield to limit cases and misunderstanding, then it is totally fine.