Asume I have an arbitrary, parametric, closed, orientable, surface; a sphere, ellipsoid, closed cylinder, weird general cone....
If you only have access to the parametrization, how can you determine whether a 3D point is inside or outside of the manifold?
I presume you are trying to determine is if a point $p$ is inside or outside the region $\Omega$ of space enclosed by the manifold, which can be seen as the boundary $\partial \Omega$ of $\Omega$. A solution is to use use Stokes–Cartan theorem: $$ \int_{\partial \Omega} \omega = \int_{\Omega} d\omega $$ with a suitable choice of the differential form $\omega$. In our case the following will work, compute: $$ I = \int_{\partial \Omega} \frac{(x-p)}{|x-p|^3} dS $$ over the manifold, where $x$ is the (3-dim) variable of integration, and $dS$ is the 2-form representing the area element of the surface. If $p$ is outside $\Omega$ then $I=0$. If $p$ is inside then $|I|$ yields the area of a unit sphere. Replace the exponent 3 in the denominator with the number of dimensions for a more general result, e.g., for $n=2$ (a curve in the 2-dim space) what you get is the winding-number of the curve around $p$ multiplied by $2\pi$.