"vector" vs "point" in definition of directional derivative Given a function $f\colon \mathbb R^n\to\mathbb R$, and given $x,v\in\mathbb R^n$, it is customary to define the "directional derivative of $f$ in the direction $v$ at the point $x$" by
$$
D_v f(x) = \lim_{h\to 0} \frac{f(x + hv) - f(x)}{h} .
$$
What bothers me about this definition is that we're calling $x$ a "point" but $v$ a "vector," even though they both live in the same vector space, $\mathbb R^n$. Is there a canonical way of distinguishing the types of $x$ and $v$, i.e., insisting that $v$ be an element of the tangent space $T_p\mathbb R^n$ (and then defining how to add a point in $\mathbb R^n$ and a vector in $T_p\mathbb R^n$), or saying that $p$ is an element of an affine space to which is associated the vector space in which $v$ lives? Or does everyone really just refer to points and vectors interchangeably in the Euclidean space $\mathbb R^n$?
 A: 
Or does everyone really just refer to points and vectors interchangeably in the Euclidean space $\Bbb R^n$?

Yes. Everyone does. 
Okay, hyperbolic statements like that one are almost guaranteed to not be true. But still, it is close to true. Practically no one balks at the concept.
However, it is also not completely true that they are used interchangebly. Both terms are used to refer to any element of $\Bbb R^n$, but which is chosen generally depends on how the element is being used. If you are just refering to the place where something is happening, then it is most commonly called a "point". If you are using the element to indicate a direction in which some is heading, possibly with some sort of magnitude also tied in, then clearly the term "vector" is to be preferred. That is obviously the case here: $x$ is the place where the derivative is being taken, so it is a "point". $v$ is the direction in which the derivative is being taken, so it is a "vector".
And as you've noted, when we generalize away from $\Bbb R^n$, it becomes necessary to separate the concepts of "point" and "vector". So in manifolds, "points" are the elements of the manifold itself (which no longer needs to be a linear space, and so the name "vector" no longer applies), while vectors are elements of the tangent space at $p$, and are used to indicate, among other things, in which direction curves are heading as they pass through $p$ and how fast they are changing with respect to their parameters. The directional derivative definition in manifolds does not depend on the adding of points and vectors together. Rather, the vectors are in a sense already "added" to the point, as they exist in the tangent space at that point.
When we then apply this generalized description back to $\Bbb R^n$, we say that the tangent space at $p$ has a "natural" or "canonical" isomophism with $\Bbb R^n$ itself, which is what we were really using in the original definition. The concept of "$x + hv$" is just the natural isomorphism that ties the element $hv \in \Bbb R^n$ with a tangent vector $w_x \in T_x\Bbb R^n$.
