Don't forget, DEFRA is there to answer all your questions! (I called them about this, too.) Don't be shy, don't feel stupid, call them whenever you're in doubt.
It DOES seem weird to get a government form from your home printer. In this case, that's how it's done. I can't remember where the serial number came from, but it wasn't our regular vet. Call DEFRA and double-check for your country/area. As written in other threads, don't trust your vet to know the process.