I'm pretty sure your second definition is an upper bound (big-O) and the first definition is a lower bound (big-Omega).
EDIT: adding a link to the wikipedia article defining Big-O and Big-Omega: http://en.wikipedia.org/wiki/Big_O_notation
EDIT: adding a link to the wikipedia article defining Big-O and Big-Omega: http://en.wikipedia.org/wiki/Big_O_notation
>It is not contradictory however, to say that the worst-case
>running time of insertion sort is [Big Omega](n^2),
>since there exists an input that causes the algorithm
>to take [Big Omega](n^2) time.
>Introduction to Algorithms, 2nd Edition, p 46