Putting this here so somebody someday will stumble on this info and will not have to run from Pillar to post like I did, for driving license history in Jeddah and Madinah/Medina.
First of all, forget getting it in Madinah, it is just not going to happen as of the date of this post.
In Jeddah, do NOT go to Dallah Driving School like people here have done before, you WILL be sent away to Muroor (Traffic Police) Headoffice in Sahafa, Aziziyah/Mishrifah. Saving grace is that Dallah driving school is really close from Sahafa Muroor Office.
At the Muroor office, go to the services area which is on the ground floor. Straight up ignore this and go to the first floor where the "Mudeer's" office is. Go to him and tell him you want PRINT.
DO NOT SAY ANYTHING ELSE, no Taqreer mufassal/mufaddal or Bayanaat Rukhsa.
The Manager may or may not ask you for a photo of a visa. I had my COPR stamped on passport, so I showed him the photo and the original license and he wrote on a yellow post-it note in Arabic "Print license - Imran Syed" and signed it.
Take this post it note down to the ground floor services hall, find the guy sitting with a dot matrix printer and surrounded by a bunch of people. Today he was sitting in front of the entrance to the services hall.
Wait for your turn, you will get a Print out on a watermarked paper.
Take this back to the first floor for the Stamp of the Manager. He will send you here and there, eventually someone will find his stamp somewhere and stamp your paper, and then he will sign it and write the Hijri and Gregorian dates.
That's it. Be happy, Go and eat Albaik.
Dallah school staff is very overworked (relatively), so prepare to be sent away as quickly as possible. Traffic Police Department staff is courteous but very sensitive, so be persistent but extra nice and careful and they'll help you out, most likely to get rid of you.