Is $\mathsf{DTIME}(n^{O(\log n)})$ an interesting complexity class?
Quoting the complexity zoo:

QP: Quasipolynomial-Time
Equals DTIME(2polylog(n)).

QPLIN: Linear Quasipolynomial-Time
Equals DTIME(nO(log n)).
Has the same relationship to QP that E does to EXP.

Both these classes exist. I know that Quasipolynomial-time algorithms arise frequently in many areas. A google search will reveal several known quasipolynomial-time algorithms. Some of these may actually be QPLIN algorithms.
One of the advantages of QP over QPLIN is that it's more robust. For example, it is closed under composition, so if you have a quasipolynomial-time subroutine that takes an input of size $n$ to an input of size $n^{\log n}$ and now run a quasipolynomial-time algorithm on this new input, you will have an algorithm that runs in $O(n^{\textrm{polylog}(n)})$ time.