We can talk about how "quickly" an infinite series approaches $0$ by talking about an asymptotic bound on its terms - a series that is $O(1/x)$ converges more slowly than one that is $O(1/x^2)$, etc.
I am confused about what we mean when we say that a function rapidly approaches a limit at a (finite) point. For example, consider the functions $x$ and $x^2$ as $x$ approaches $0$. I think we say that $x^2$ approaches $0$ "more quickly" than $x$ does as $x\to 0$ because $x^2$ is $o(x)$ as $x\to 0$, so $x^2$ reaches any arbitrary "closeness to $0$" before $x$ does as $x\to 0$.
On the other hand, close to $0$, it also seems that $x$ is approaching $0$ "more quickly" than $x^2$ in the sense that the magnitude of derivative of $x$ is larger than that of $x^2$.
So my question is which one of these does "more quickly" generally mean in informal math-language, and if anyone has any insight into the intuition here.