An inverse limit is the same thing as a limit. (Similarly, a direct limit is the same thing as a colimit.) In this context, an inverse system is the same thing as a diagram, and an inverse cone is the same thing as a cone.

Many authors restrict this terminology to limits over codirected sets (or cofiltered categories); see codirected limit (or cofiltered limit) for discussion of this case if you think that it may be what you want, or see limit for the more general notion. Especially common is the codirected set $(\mathbb{N},\geq)$ of natural numbers, in which case the inverse system may be called an inverse sequence.