Note that a bicartesian closed category is bicartesian (that is, it is both cartesian and cocartesian), and furthermore it is cartesian closed, but it is usually notcocartesian closed (as the only such category is the trivial terminal category), nor co-(cartesian closed) (i.e., the dual of a cartesian closed category; aka, cocartesian coclosed). Thus the terminology could be confusing, but since the only categories which are both cartesian closed and co-(cartesian closed) are preorders, there is not much danger.

Also note that a bicartesian closed category is automatically a distributive category. This follows since the functors $X\mapsto A\times X$ have right adjoints (by closedness), so they preserve colimits.

A bicartesian closed category is one kind of 2-rig.