Analysing worst-case time complexity of quick-sort in different cases
The usual rule here is only one question per post.

Link [2] doesn't say that random pivot and median pivot lead to $O(n^2)$ time complexity.

The median of first, middle, and last element is not the median of the whole array.  Writing "median pivot" is probably a bad idea, unless you have carefully specified what that's a median of.

Link [1] says that taking the median of all elements leads to $O(n \log n)$ time complexity.  That's different than taking the median of first, middle, and last.  Also, it's rare for practical implementations to take the median of all elements, as that link already states.