The Divergence Theorem is generally defined as
\begin{align} \int_\mathcal{V} (\nabla \cdot \vec F)\mathrm dV = \oint_{\mathcal{S}}( \vec F \cdot \vec n)\mathrm dS. \end{align}
Moreover, the classical Stokes Theorem is generally defined as
\begin{align} \int_\mathcal{S} (\nabla \times \vec{F}) \cdot \mathrm{d} \mathrm{S} = \oint_{\mathcal{L}} \vec{F} \cdot \mathrm{d}\mathrm{L}. \end{align}
Here $\oint_{S}$, $\oint_{\mathcal{L}}$ are defined as boundary or closed surface/path integral. In the literature a closed surface/path is generally defined as one closed path/surface.
Considering a 2D annulus shape, e.g., a circle with a hole, there is not only one closed path, rather we have two closed paths.
From a formal and mathematical point of view, is it reasonable to distinguish between those two differently closed integrals in a similar way of either writing $\int$ or $\oint$.
Edit:
From the discussion below we may summarize, that a distinction between both integrals does not always makes sense, e.g., Stokes theorem, however considering the Divergence theorem it might be reasonable.