Weak (graph) homomorphisms are mappings $f: V(G) \rightarrow V(G')$ such that the images of connected nodes $x,y$ (in the source graph) are connected:
$$R(x,y) \rightarrow R(f(x),f(y)) = R(x',y')$$
Strong (graph) homomorphisms are mappings $f: V(G) \rightarrow V(G')$ such that the pre-images of connected nodes $x',y'$ (in the target graph) do exist and are connected:
$$R(x',y') \rightarrow R(f^{-1}(x'),f^{-1}(y')) = R(x,y)$$
I wonder why "medium-strong" (graph) homomorphisms don't play a more prominent role, which one can rather simply define as mappings $f: V(G) \rightarrow V(G')$ such that the pre-images of connected nodes $x',y'$ are connected if they exist, which is (I hope) equivalent with:
$$\neg R(x,y) \rightarrow \neg R(f(x),f(y)) = \neg R(x',y')$$