What does a faster algorithm mean in theoretical computer science?
No, an algorithm running in time $O(f(n)/g(n))$, where $g(n) = o(f(n))$, is not necessarily considered an improvement. For example, suppose that $f(n) = n$ and $g(n) = 1/n$. Then $O(f(n)/g(n)) = O(n^2)$ is a worse time bound than $O(f(n)) = O(n)$.

In order to improve upon an algorithm running in time $f(n)$, you need to come up with an algorithm running in time $o(f(n))$, that is, in time $g(n)$ for some function $g(n) = o(f(n))$.

If all you know is that an algorithm runs in time $O(f(n))$, then it is not clear whether an algorithm running in time $O(g(n))$ is an improvement, whatever $f(n),g(n)$ are. This is because big O is only an upper bound on the running time. Instead, it is common to consider the worst-case time complexity, and to estimate it as a big $\Theta$ rather than just as a big $O$.