Parallelly transported vector field

Suppose a manifold $M$ with a covariant derivative operator $\mathrm{\nabla }$ defined on it (affine manifold). A vector field $X$ on $M$ is a parallelly transported vector field along a curve $c$ if

It is also said that the vector field is constant along the curve.
This is well defined since covariant derivative only depends on the value at a point in the first vector field (see linear connection#Definition as operator).

If the vector field $X(t)$ is only defined along the curve $c$, we consider the covariant derivative along a curve $D_c$. With this in mind we can say when a vector field defined along a curve $c$ is parallel along the curve: $D_{c}X=0$ for every $c(t)$.

Weaker notion: parallel vector field

Given an affine manifold, a vector field $X$ is a parallel vector field with respect to a curve $c$ if

for a certain smooth function $\mu :\mathbb{R}\to \mathbb{R}$.