P versus NP and what we are looking for
P vs NP is settled as soon as we find a $O(n^k)$ algorithm for an NP-complete problem (for any constant $k$) or show that no NP-complete problem has such an algorithm. 

Whether or not $k$ is too big to yield a tractable algorithm is a different problem. Having a practical algorithm for, say, SAT, would indeed be very useful, many problems we currently solve heuristically (e.g. in chip design) could then be solved to optimality.