First of all, it doesn't make much difference how you conceive of or draw
the vectors when the manifold of interest is globally flat, since the
manifold itself and its tangent space to it at each point are isomorphic.
In such a case we can treat vectors in the usual way as having them reside
in the space of interest (i.e. R^3).
Things only become more complicated than this when our 'space', i.e. our
manifold of interest, is curved.
Regarding the specific question above, in the general curved space case,
the position of a point in that curved manifold is *not* a vector at all.
It is a point in the manifold. The points in a manifold are not
individually vectors. Vectors live in vector spaces; curved manifolds
are not vector spaces. Since the position is not a vector, it doesn't
reside as a vector anywhere. It just resides as a point just where it
is in the manifold.
*But*, the *differential displacement* along a path between two
infinitesimally nearby points in the manifold *is* a vector, and it lives
in the tangent (vector) space of the point where that differential is
taken.
If you want to visualize the tangent space that is tangent to the
manifold at some point, you can imagine the origin of the tangent space's
vector space (i.e. its zero vector) as coinciding with the point of the
manifold about which the tangent space is taken. The basis vectors of
this tangent space are (understandably) tangent to the manifold locally
in an infinitesimal neighborhood of the point of interest, and point away
from it along each basis direction. At finite distances away from the
point of interest the manifold curves away from the tangent space and is
no longer tangent to it that far out.