In most cases where the word “cofinal” is used, it seems to be the case is a subdiagram of in whatever sense of subdiagram appears suitable. In that case the cofinality is equivalent to the inclusion functor being a cofinal functor.
As defined above, no relation is posited between and , and so it seems not too evil to define cofinalness1 as “a single object is a colimit for both diagrams”.
It is not said that they are final if they have equivalent limits — the “co”s are not freely mutable, although the dual situation is doubtless just as interesting.
Let , and let be an initial cocone over ; it seems natural to ask that be an initial cocone over .
Nonempty subsets of a finite total order are cofinal iff they have the same maximum.
Every infinite subset of is cofinal with , as diagrams in .
It is often said that two diagrams are cofinal even when neither has a colimit, if they acquire a common colimit on passing to a suitable completion of . This can probably be phrased internally to , at the cost of intuition.