"Region" seems to have two competing definitions in complex analysis — which one did you learn?

I'm reading up on "point sets" in complex analysis and I have stumbled upon two different definitions of the term "region", see below: Version 1. A region is a domain together with none, some or all of its boundary points. Version 2. A region is just another word for "domain". my textbook is using the first version while sources like Wolfram and Wikipedia is using the second version. But while reading more about it, there also seems to be a split across different authors and languages which one that is used. Have anyone looked into this before and can share the history of how this come to be and if there is a more "correct" one between them? //