An undirected graph $G$ is biconnected if when removing any vertex, $G$ remains connected. It is $k$-connected if when removing any $k-1$ vertices, $G$ remains connected.
In the literature of graph theory when people say $G$ is biconnected, is it implicitly mean that $G$ can be also $k$-connected graph for $k\geq 2$?
I am asking this because when looking to the ways of computing st-numberings of a graph, all the work assume a biconnected graph.
According to your definition, biconnected is equivalent to $2$-connected. If $G$ is $k$-connected for $k \geq 2$, then it is indeed biconnected. However, the converse does not hold. You can have a biconnected graph that is not $k$-connected for $k \geq 2$.