What's the difference between "adaptive sorting" and "sorting almost sorted data quickly"?
I think the importance of "sorting almost sorted data quickly" in the database community is because of good streaming and/or parallel algorithms for k-sorting and for k-sorted id generation. (e.g. http://link.springer.com/chapter/10.1007%2F978-3-540-77050-3_2#page-1) Adaptive sorting is a more general topic: for instance, reverse order would not be what the database community cares about, but would be perfectly reasonable to exploit more generally.