What Unity says about points vs. vectors is meaningless in the long run, because geometry APIs just pick distinct definitions to make the tool more accessible, they don't correspond to how these things are conceptualized in geometry. Take a look at the implementations of the classes, if you can. Because it's arbitrary, to know its definition is the only way to understand what the concept is. Full disclosure, I don't have Unity experience.
A vector is a point in a vector space, in that the concept of a point in the geometry is encoded by elements of the underlying set. A vector space has a distinguished vector, called the origin or 0. Linear algebra is an attempt to encode a fragment of euclidean geometry w/ an origin algebraically.
The arrow and its length
Motions across a space of points are frequently interpreted as all the arrows from source/before points to their target/after points.
A function of two arguments can be applied to one argument to produce a function of one argument - we can speak of x+, the function which takes each vector y to the vector x+y. This is the translation associated w/ adding x. The associated arrows run from points y to points x+y. See: partial application, currying.
So why do we only use the one arrow? The arrow from the origin points to a specific vector, the x in x+ - the origin is the identity of vector addition. So, we can recover the translation x+ from just its value x+0 = x.
As a graphical representation of the space, the arrow representation has to do with our ability to visually or physically extrapolate the effect of a translation from the value determining it. When do we have that ability?
To give the vector space a norm making it a normed vector space is to provide a notion of the length of a vector that makes sense as its distance from 0. As well, this is to be a distance satisfying the triangle inequality, which is a strong constraint on how the lengths of two vectors relate to that of their sum. From length we can define distance to make this a metric space, and a geodesic is a path that's intrinsically straight in that it's as short as possible. The euclidean norm induces euclidean distance and the geodesics are the line segments of the arrows, but if you draw the arrows as geodesics using different norms, you could extrapolate the geometric effect of the translation from the geodesics to learn about the geometry.
The meaning of point and vector
In some cases in doing games geometry, your space of points is not a vector space. An affine space of dimension n can be embedded in a projective space of dimension n. Affine maps reduce to projectivities. Projectivities also let you do FOV, w/c I think is not affine. Projectivities have benefits:
The projective n-space over a field can be constructed from the linear (n+1)-space (vector space), by treating the points of the projective space as the lines through the origin of the linear space. Planes through the origin in turn give projective lines. Multiplying vectors by a fixed matrix is a linear map, this is what matrix multiplication is for. Linear maps preserve the origin and are compatible w/ incidence. In particular, if f is a linear automorphism (corresponding to an invertible (n+1)x(n+1) matrix), and two lines L, M through the origin span a plane A, then f L, f M are lines through the origin spanning f A, so f will preserve incidence on the projective space as well - an invertible matrix has an associated projectivity. Matrix multiplication encodes composition of linear maps, and hence of the projectivities.
Removing the origin from the linear space, all the points on a given line through the origin are scalar multiples of one another. Exploiting this fact, homogenization picks a linear point to stand in for each projective point and an invertible matrix to stand in for each projective transformation (as in this 2D -> 2D affine maps as 3D -> 3D linear maps video), in such a way that the representatives are closed under matrix-matrix and matrix-vector products and give and are given by unique projective things. This description of construction of the projective plane from the linear plane ties some things together.
So, in the model-view-projection matrix pipeline, we're using vectors to represent the points of our projective space, but the projective space is not a vector space, and not all vectors in the vector space we're using represent points of our geometry (see picture of affine plane at right). We use translation matrices instead of vector sum if we want translations. Sometimes, people call projective or affine points vectors, especially when using a setup in this vein.