I have more persuasive evidence as to why the real shooting of Oswald was in the PD and not in the ambulance.
First, note that it was important that they get Oswald shot as soon as possible, in the hope that he would either die before reaching the hospital, which was their preference, OR that he be in hopeless, unsavable condition by the time he got there, which was the case. If he had survived, it would have been an absolute nightmare for them. So obviously, on that basis, it was in their interest to shoot him at the PD and not the ambulance.
And here is something concrete along that line:
We know that they made an unnecessary circle that wasted several minutes of time getting to the hospital. All they had to do was turn right as they exited the Commerce Street ramp and gone the wrong way on a one-way street for like 200 feet to the corner and then turned right on Harwood. That was all the "against traffic" driving they had to do, and by then, the area was crawling with police who could easily control traffic and clear the way. It would have been easy as pie. So, why did they go the way they did? To kill time. To give Oswald more time to die. And it worked. Even though he didn't die in the car, by the time he reached the hospital, he was beyond saving and beyond hope.
Now, why would they do that if Oswald wasn't shot yet? The death- clock only starts ticking AFTER you are shot, not before. So, it would have been pointless to do it if Osawld hadn't yet been shot. Were that true, his death-clock wasn't even ticking yet. Ipso facto, they must have shot him in the PD.
And note that the ambulance driver said that it was the cops who told him to go that way. Of course, his WC examiner had no follow-up and no problem with it.
Shooting Oswald on the way to the hospital would have been way too risky because Oswald could have survived, and if he had, it would have been all over for them. Imagine what his lawyer would have done when he suffered such an attack when in the custody of police. As it was, Marina Oswald should have filed a wrongful death lawsuit against the Dallas Police based on negligence. But, she, of course, is a whole other story. But, if Oswald had survived, you can be sure his lawyer would have done it- and more.
Furthermore, how could they attack him again? They could hardly set up another "there was nothing we could do" scenario. Who would believe it? That he was attacked twice in police custody?
So, they had to make absolutely sure that he died from this attack, and the sooner they got that bullet into him the better.
I'm sure the risk of dying increases logarithmically with the passage of time. So, every minute, in fact, every second that they could delay his getting medical and surgical attention was invaluable. But, it only counted if he was already shot. So, Oswald was shot in the PD. Not in the garage, but in the building after the spectacle.