Dual/complement of independence system
The usual terms I know from the theory of posets and lattices are downset and upset (or upper set). Also as the empty set is a subset of every set, I think your first condition is vacuous as long as the system is not empty. Same for the first condition in the second definition.

If a downset is closed under joins (in your case set unions), it is called an ideal. If an upset is closed under meets (in your case set intersections), it is called a filter.