I'm glad to see your concern for over-vaccination, that is a real concern, most people don't give it much thought.....
I'm sure you'll get many opinions on this situation so here's mine.
....The pup is a little over 12 weeks old and you don't know if he had every vaccine under the sun or just one....This is from Whole Dog Journal....
In the August 2008 issue of The Whole Dog Journal, Dr. Ronald Schultz reports in an article entitled, Vaccinations 101, by Lisa Rodier, "Research shows that less than 50 percent of puppies will respond at six weeks; 75 percent at nine weeks; 90 percent at 12 weeks; and by 14 to16 weeks, close to 100 percent will respond. "
From this observation your pup had somewhere between a 75% to 90% chance what ever vaccines he had "took". But which one did he have??
From what I've read the dogs system does not fully mature until they are about 6 months old. That is the ideal time to do a vaccine. BUT, most people don't want to wait that long so in they go. If a person waits until the pup is older then all they need is the one shot, not the three people are usually told to do. You could follow the new protocol for vaccines that is in effect and just do the "core" vaccines which would be distemper, parvo, and rabies.....Dr Dodds has a schedule here...
Vaccination Schedule Recommendations For Dogs
When you go to the vet only allow one vaccine per visit, NEVER allow a combo vaccine...NEVER allow a rabies to be done on the same visit as another vaccine, it should be done on it's own visit. Always space the vaccines 4 weeks apart from each other.
Here are some good sites to read:
Vaccinating Dogs: What Your Vet Hasn't Told You Too Many Dog Shots: Eliminate Unnecessary Vaccines and Improve Health | Truth4Dogs Puppy Shots
Good luck...