In the literature this sort of equivalence is often called a biequivalence, as it has traditionally been associated with bicategories, the standard algebraic definition of weak -category. There is a stricter notion of equivalence for strict -categories, which traditionally is called just a -equivalence and which on the nLab is called a strict 2-equivalence.
A 2-functor can be made into part of an equivalence iff it is essentially surjective on objects, essentially full on 1-cells (i.e. essentially surjective on Hom-categories), and fully faithful on 2-cells.
Just as the notion of equivalence of categories can be internalized in any -category, the notion of equivalence for -categories can be internalized in any -category in a straightforward way. The version above for -categories then results from specializing this general definition to the (weak) -category of -categories, (weak) -functors, pseudonatural transformations, and modifications.
There is one warning to keep in mind here. Every -category is equivalent to a semi-strict sort of -category called a Gray-category, since it is a category enriched over the monoidal category Gray of strict -categories and strict -functors. Of course itself is a Gray-category, but as such it is not equivalent to the weak -category of weak -categories and weak -functors.
In particular, an “internal (bi)equivalence” in consists of strict -functors together with pseudonatural equivalences relating and to identities. This is a semistrict notion of equivalence, intermediate between the fully weak notion and the fully strict one.