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This is a tricky one. The easiest way to think of it is one runner is out attempting to advance on a wild pitch/pass ball while another runner advances safely.
The reason they don't simply score it as "advanced on wild pitch/pass ball" is because the official MLB rules* state that if an out occurs on what would normally be ruled as a wild pitch or pass ball (throwing someone out at home who was trying to score on a ball that got away from the catcher, for example), then it's not actually a wild pitch/pass ball because an out was made on the play. The other runners who advance safely on the play are said to have done so on a "runner's fielder's choice."
*See the official comment for MLB rule 10.13 (Wild Pitches And Passed Balls).
Yes, it counts as an 0 for 1. The reasoning behind it, is this: A fielder's choice usually happens when there is a runner on first, and you hit a grounder. The fielder could easily throw you out at first, but he CHOOSES to get the runner out at 2nd instead. You effectively have made an out, and not reached base safely via a hit, but the only reason you're on base and not the other guy is because to the fielder, it was more advantageous to get the lead baserunner out.