Ship's manifests were made out in port before the ship left. I've seen cross-outs before. They happen when someone buys tickets but then doesn't actually make it on the boat (could be due to illness). Hopefully, the boat where your relatives' names were not crossed out was the last of the three.
Another possibility is clerical error. If the clerk had several manifests on his (likely) desk and just wrote in the wrong one.
Yes, the one where they were not crossed out is almost certainly the one they traveled on.