$G$ is $2$-vertex-connected graph if and only if every $2$ vertices from $G$ lie on a simple cycle.
If $G$ is $2$-vertex connected graph it means for every $ v \in V(G), \deg(v) \ge 2 $ because if exist any $ v_1 $ and $ \deg v_1 = 1 $ then we can erase neighboring vertex and graph will not connected so won't be 2-vertex connected. So if $ v \in V(G), \deg(v) \le 2 $ then it exist cycle. $ \deg_v \ge 2 $ so if we start in $v$ we have to back to this vertex.
$\impliedby$ I don't know.