I don't know a name, but I'll give you a different characterization. Biconnectivity is sufficient but too strong, while "having minimum degree at least 2" is necessary but too weak. I think this is a necessary and sufficient condition:
$G$ has minimum degree at least 2, and if v is a cutvertex of $G$, then there is some new connected component of $G - v$ with at least two vertices adjacent to v.
This is equivalent to: Removing any vertex of degree d increases the total number of connected components by at most $d-2$. Some generalization of this property may have a name.