I know this is irrelevant to the larger point of your post, and I'm sure that you know this already, but the worst case is the big-O. This is just another reason why "big-O" is not the most helpful thing to discuss in practice.
Big-O is a way of categorizing the growth of mathematical functions. Those functions can represent anything. It is wrong to talk about the big-O of an algorithm without specifying what you are measuring. Be it average operations, worst case operations, average memory, worst case memory, amortized average time, amortized average memory, and so on.
It happens to be that when we talk informally, we're usually talking about the worst case we are willing to think about. Quicksort's worst case is a sorted set, so we think of that as O(n^2). But then we turn around and cheerfully call a hash lookup O(1) because its O(n) worst case is incredibly rare in practice.
Thanks for correcting me!
I thought you were right at first but then realized what was going on. This is a pretty subtle point and mostly uninteresting for well-understood algorithms like quicksort. But one slightly less subtle point is that big-theta isn't average case, it is the combination of big-O and big-omega, i.e., bounded from above and below (possibly with different constant factors) by the same asymptotic behavior.