That's incorrect. The /mathematical/ definition means that it's the upper bound. I.e. yes, the set of all functions of O(N) is a subset of the set of all functions O(N^2). So you could say Quicksort is O(N^100) and still be technically correct.

However, the big-O notation does NOT specify anything about worst/avg/best-case complexity of a given algorithm. That should still be defined in the analysis.

You mixed up those two slightly different concepts.