Is there a word for the status of a team being 'home' or 'away'

Solution 1:

Despite you ruling it out, venue seems to be most commonly used even when there is no particular venue associated with it.

If you look at the Home Advantage page on Wikipedia, which falls into the Terminology Used in Multiple Sports category, it repeatedly uses "venue" when talking about the location of a sports game. Neutral Venue also redirects to this page.

Solution 2:

I suppose what you are looking for would be the name of the enumeration of home, away, and possibly neutral. In sports you could probably use the term berth since this is a term used for sports tournaments already, as in this team has earned a berth in the finals. If you search Google for "earned a berth" you will find multiple headlines using the word this way.

If we extend this idea to mean that there are two berths in a final, you could say that one berth is the home berth, and one is the away berth.