You really should have posted this in the medical forum so that the medical people would see it.
It is very common for puppies to have intestinal parasites. It is also very common for puppies to have parasites when the breeder says they already wormed them.
What is the name of the parasite that your vet found on the stool sample test? From the description of the medication, giving it for 10 days, it sounds like it might be Coccidia. What doesn't match up is the worm he vomited. There are no adult stage worms you will see with Coccidia, but there is with Roundworms. I would call your vet and tell them about the one he vomited up and ask the vet if that makes sense, and if he was wormed for roundworms.
I would think that your vet would have told you to keep the dogs completely separate from each other, but the most important thing is to be sure your old dog does not come in contact with any stools of the new pup. That is how it is transmitted usually.
Be sure to get another stool sample test done at the end of the medication to be sure the pup is parasite free. It is not uncommon to have to do another round of the medication.