That is correct. Presidential transportation changed drastically following the assassination of John Kennedy in Dallas. Up to that time, the use of the bullet resistant bubble top was up to the desires of the President. Following Kennedy's death, the Secret Service determined that transporting the president in an open top car would be the rare exception in the future. The President always travels in the Secret Service limo for security reasons and there is always a backup car on hand. Construction of the car is classified information but it is bullet resistant and bomb proof. These cars are operated by the Secret Service and are stored in a secure facility in the DC area when not in use. Similar limousines are kept at the Secret Service training facility for training and refresher purposes. One ex-presidential limo, the X-100, is owned by the Henry Ford Museum. That is the car in which Kennedy died and was modified later with a permanent hard top.
In short, the reason that the President travels in his personal limo is that for security reasons, the Secret Service will not risk their protectee's safety by allowing him to travel in a vehicle that they do not have full control over.
"When a man runs on railroads over half of his lifetime he is fit for nothing else-and at times he don't know that."