What does it mean for a function to "quickly" approach $0$? 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.

Notice that you are not using the size of the derivative in discussing limits at $\infty.$ So, you don't emphasize the derivative in limits at $0.$ You just talk about which function gets close to $0$ earlier, which at $0^+$ means for larger $x.$ Also, at $\infty^+,$ the derivative information (in absolute value) agrees with the size information; note that we are thinking of a point moving to the right along the $x$ axis.
I suppose it is fair to say that the derivative information is backwards of this, at $0^+, $ and this is happening because approaching $0^+$ we are moving to the left. The slowest approach to $0$ you are likely to see is Holder continuity such as $\sqrt x,$ where the derivative is actually infinite at the origin. `Slow' meaning: you need to have $x < 0.0001$ to get $\sqrt x < 0.01$
