Are there variations of the regular runtimes of the Big-O-Notation?

Question

There are multiple $O$-Notations, like $O(n)$ or $O(n^2)$ and so on. I was wondering, if there are variations of those in reality such as $O(2n^2)$ or $O(\log n^2)$, or if those are mathematically incorrect.

Or would it be a right thing to say that it is possible to improve a $O(5n^2)$ to a $O(3n^2)$? I can't and don't need to figure out runtimes yet and I do not need to improve anything, but I'd need to know if this how you describe your functions in reality.