What you call a cut position for your grid word is called a cut vertex or articulation point in general graphs. From Wikipedia:
Specifically, a cut vertex is any vertex whose removal increases the number of connected components.
And a bit further down in the same article:
The classic sequential algorithm for computing biconnected components in a connected undirected graph due to John Hopcroft and Robert Tarjan (1973) [1] runs in linear time, and is based on depth-first search. This algorithm is also outlined as Problem 22-2 of Introduction to Algorithms (both 2nd and 3rd editions).
Having determined the biconnected components, it should be quite easy to determine the articulation points: All nodes which are contained in more than one bi-connected component are articulation points.