Of course, using the most abstract definition is not the best idea, but anything equivalent to the standard terminology is ok. For example, in 2d, we can san say the convex hull of a finite set of points is the smallest enclosing convex polygon.

The name convex hull is related to linear hull and conic hull, therefore defining it to not contain the interior is strange, but reasonable because in many applications we only care about the description of the convex hull by describing the polygon.