Is there an official definition of a 'runner's fielder's choice'?

Answer 1

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).