In U.S. sports (and sports reporting), games are for the most part identified as being home games or away games, but that simple and seemingly exhaustive split doesn't cover all of the venue possibilities in detail. At least five possibilities exist:
Team A plays Team B on Team A's regular home field.
Team A plays Team B on a different field in Team A's home region.
Team A plays Team B on Team B's regular home field.
Team A plays Team B on a different field in Team B's home region.
Team plays Team B at a truly neutral site at which neither side can claim a home-field advantage.
Under standard U.S. parlance, situations 1 and 2 are classified as "home games," situations 3 and 4 are classified as "away games," and situation 5 is classified as a "neutral-site game." You could argue that situation 5 involves what amounts to an away game for both teams—but for purposes of grading the quality of the victory by the winning side, it makes sense to distinguish it from (on the one hand) a victory achieved in front of a strongly supportive crowd and (on the other) a victory achieved in front of a passionately hostile crowd.
Since the status that the OP seeks a name for changes with each game, there is no reason why the OP couldn't replace it in the outline example with the word game, accompanied (for each particular game) by one of the three values "home," "away," or "neutral site."
However, I think that the word site, plus one of the three values "home," "away," or "neutral" works even better.