DOmain analysis - why OFF points are inside of the domain when the border is open

I have asked on a few other sites, no response but it must be something silly as many authors mention in their books.

Here is the best text I found:

My ultimate question is: 1Why the OFF point lies INSIDE the domain when the border is open. If it is about being off the border, why cant it be outside too?

For closed domain, both OFF and ON points are outside domain, while for open domain, the OFF point is in inside the domain. I just do not get why is that.