What is the exact definition of the tension of a rope?

Here http://web.mit.edu/8.01t/www/materials/modules/chapter08.pdf the following definition is given

> The tension   $T(x)$ in a rope at a distance  $x$ from one end of the rope
> is the magnitude of the action -reaction pair of forces acting at the
> point  $x$

But I think that this is valid only in the case that the tension is the same, throughout all the rope. Is that definition still valid in that case?