> Yay, bikeshedding. “edge” seems fine to me, but “threshold” less so. It makes me think of crossing doors and gates into something else. (I.e. it has the same problem for me that you had with “border”.) Perhaps “limit” or “boundary”. (You may feel “boundary” is pretty much “border”, but I see it more as the limit, rather than an area with action on either side.)
I should mention that boundary, in mathematics, is a set of points (not just one point), and the boundary of a set is not necessarily within the set. So the use of "boundary" does not seem appropriate to denote one point which is always in that set. Some researchers use "frontier" for the same purpose, and both "border" and "edge" are close enough to "boundary" in their colloquial use; I would say all of these are not very good names.
I do not know why I dislike "threshold", it just sounds weird in this context, but I am not a native speaker.
The term that I like would be the "terminator" (as used in astronomy). Or, perhaps, the "terminus" or the "terminal".