Integral curves are defined in Wikipedia by the formula
$$\mathbf{x}'(t) = \mathbf{F}(\mathbf{x}(t)).$$
What about a modified definition such that requires only that the vectors $\mathbf{x}'(t)$ and $\mathbf{F}(\mathbf{x}(t))$ are of the same direction (not necessarily the same length)?
I am almost sure that there is some (widespread) term for such modified definition of integral curves.
(Regular) integral curves for a vector field seem to correspond bijectively to such modified integral curves for a slope field. I asked about this in another question.
In the comments I suggested "reparameterized integral curves" as a name for these things. The only question is whether such things really ARE reparameterized integral curves. Here's a quick proof that they are (under very mild constraints):
Suppose that $x'(t) = c(t) F(x(t))$ for some function $c$ that's everywhere positive. (I assume that this is included in your notion of "pointing the same direction".
Let
$$y(u(t)) = x(t),$$ where $$ u(t) = \int_0^t c(s) ~ds, $$ which, because $c$ is positive, is a monotone function, and therefore has an inverse defined on its image.
Then \begin{align} y'(u(t)) u'(t) &= x'(t) \\ &= c(t) F'(x(t)) \text{, so}\\ y'(u(t)) c(t) &= c(t) F'(y(u(t))) \\ y'(u(t)) &= F'(y(u(t))) \end{align} and letting $s = u(t)$ we have $$ y'(s) = F'(y(s)), $$ (at least for $s$ in the image of $u$) as needed.