In the case of fields, it is clear what a local transformation in the internal space of the field is:
$$\phi(x) \to \phi'(x')= G(x) \phi(x),$$
as opposed to a global transformation, where $G$ would not depend on $x$.
But I don't understand the difference between local and global coordinate transformations.
It is said that general coordinate transformations, $$x^\mu \to x'^\mu =x^\mu+\epsilon^\mu(x)$$ are local, while rotations $$x^\mu \to x'^\mu =x^\mu+\sigma^\mu_\nu x^\nu$$ are global.
How can we distinguish between the two? In both cases, the variation depends on $x$.