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Runner on 2nd. Batter hits a soft grounder to the pitcher. Pitcher unsuccessfully attempts to look back the runner at second. Runner maintains their leadoff and the pitcher throws the ball to 2nd baseman and a rundown play ensues. The batter arrives safely at first. After a few throws back and forth, the 3rd baseman throws the ball to 2nd base as the runner is retreating back. The ball is partially deflected off the runner and rolls into the outfield. The runner on 2nd base goes to 3rd and arrives safely (The outfielder eventually got the ball and made a throw to third that arrived a moment late). The batter moved up to 2nd when the outfielder threw to third.

I currently have it scored as the batter arriving at 1st on a fielders choice and then advancing to second on an error by the third baseman. The runner on second also advancing to third on the same error by the third baseman.

An alternative I was considering was batter safe at 2nd on a fielders choice and the runner at second advancing to 3rd on an error by the third baseman.

Or batter safe at first on fielder choice, advancing to 2nd on the throw. Runner on 2nd base moves to 3rd on an error by 3rd baseman.

What is the proper way to score this strange play, and how do you enter it in iscore?

I would pick the 3rd option based on your description.

Had the runner not been hit by the throw from 3rd would he have been safe at second?

The runner was still about three steps from 2nd, so applying the tag would have been very likely, but not assured. Had the runners not advanced I wouldn't have charged an error to the third baseman on the play.

In play
FC
Locate
Order of touches 1-6-5-4
What happened to runner on 2nd, tap third, error, 5
What happened to batter, tap second, on the throw

I'd go with the first or third option, based on your description of the play. And iScore will only count it as one error anyway if you choose the first. Had the error not been made, neither runner would have advanced, but you could specify the "on the throw" all the same.