Ok for those who don’t like abstractions, here is a simple concrete illustration.

TeamA plays TeamB, five game series. Their home fields are FieldA and FieldB.

TeamA odds of winning any particular game (according to Las Vegas) are 35% if played at FieldA and 25% at FieldB. (Nevermind that there can be different odds for different games, for example for different starters; you just have to come up with a more complex equation but the conclusion will be the same. This is simplification without loss of generalization.) If home field is to be FieldA for games 1, 2, and 5 (if necessary), then TeamA has home field advantage (even though TeamB is favored overall). If they are to be played at FieldB then TeamB has home field advantage.

If the odds of TeamA winning any particular game (according to Vegas) is 35% no matter whether at FieldA or FieldB, then there is no home field advantage, for either team, no matter where played.

Let’s say the odds for TeamA are 25% for games at FieldA and 35% at FieldB and games 1 ,2, and 5 (if necessary) are to be played at FieldA. Then TeamA has a home field DISadvantage.