There are (at least) two definitions for rigidity of a local system on $X = \mathbb{P}^1\setminus\{p_1,\dots,p_n\}$:

- A local system $L$ on $X$ is rigid if any other local system with conjugate monodromy around each of the $p_i$ is isomorphic to $L$
- A local system $L$ of rank $n$ on $X$ is rigid its preimage under the map $Hom(\pi_1(X),GL(n))\to M(X,n)$ is open.

In the literature people seem to treat these definitions as being equivalent, but I don't see how they could be. For example rank $1$ local systems are clearly always rigid in the first sense. But the map $Hom(\pi_1(X),GL(1))\to M(X,1)$ is an isomorphism, so the preimage of a (closed) point will never be open.

So what is the relation between these two notions of ridigity?