There is a certain bit of controversy on this since at the time of the play, both runners were forced to the next base. My interpretation is as you have stated.
If the defense does the appeals in the wrong order, that is, if they appeal the out at 2nd before the out at 3rd, then the appeal at 3rd is no longer a "force" play and the run would score. Just as the defense must tag a runner forced to vacate a base before they tag the base, they must appeal the forced runners in the proper order to retain the "force".
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