The modified performic uses a LOT of DCM, which is immiscible with the aqueous phase, and the "original" uses just a tiny bit of acetone, which is miscible with the aqueous phase.
If there wasn't so much formic acid as solvent in the original procedure, the isosafrole and/or ketone would just oil out from the solution, killing the yield. In the modified procedure with DCM, the isosafrole/ketone is always dissolved in the organic phase, and when the performic acid is added, it too will dissolve in the DCM, but not the water the solution contains.