Does big-Oh notation in optimization follow the same convention as in CS?
They are using the standard Landau notation.

In the theorem you stated, it is said that after $k$ steps your distance from the optimal value is smaller than $\frac{c}{k}$, where $c$ is a constant which depends on your initial distance from the point where optimum is achieved, and on the step size. In that case you can indeed say that the convergence rate is $O\left(\frac{1}{k}\right)$, which is interpreted as you quoted in the following line, to get $\epsilon$-close to the optimal value one needs $O\left(\frac{1}{\epsilon}\right)$ iterations.

Note that I wouldn't take this answer as "anywhere in optimization this notation means the Landau notation", and you are probably right to say that it needs to be mentioned. However, in this context, you can pretty safely assume that this was the authors intention.