Baseball is one of the rare sports where a home-field advantage is actually written into the rules (hockey is another one). Batting in the bottom half of an inning ALWAYS give the home team the advantage.
Baseball doesn't have sudden-death extra innings. Both teams get to bat the same number of times regardless of how many runs are scored in the top half of an inning (unless the home team is leading after the top half of the ninth inning).
This advantage is magnified in extra innings, because in the bottom half of the inning the home team knows exactly how many runs they need to score in order to win the game.
Look at the scenario presented here. It's based on an implicit assumption that the visiting team scored no runs in the top half of the inning -- because it would be ludicrous for the home team to give up two outs to score one run if the visiting team has already scored (for example) two or three runs in the top half of the inning.
That’s why my post also included:
“Granted, the home team gets to bat in the bottom of the 10th, but the visitor runs may have already been scored.”
And yes, we all understand home field advantage.