It appears to me that you are scoring the play the same way regardless of when the runner at third breaks for home. You have to score the play, IMO, depending on when the runner at third breaks for home. If the runner breaks for home before the catcher throws to first and the catcher chooses to get the runner at first instead of the runner at home then that I would score the hitter with a strikeout and the out at first, 2-3, catcher to first base. The runner at third advances and scores by defensive indifference. The defense allowed her to score without a play at the plate. Same thing basically as a runner advancing from first to second without a throw from the catcher.
If the runner at third does not break for home until the catcher throws to first, then that would not be defensive indifference. The hitter would be out on a strikeout and, 2-3, throw from catcher to first base. Scoring the runner from third would be a judgement call by the official scorekeeper. Could score it a stolen base are just simply the runner advancing on the throw to first. Again just my opinion as an official scorekeeper for over 50 years.