Hey David! I have to say that working in rehabilitation is awesome career. I think you'll really like it. Here's a short answer to your question:
(PT) Physical Therapists go to SEVEN years of school: 4 years for a general BS + 3 years for a Doctorate in Physical Therapy
(PTA) Physical Therapist Assistants go to TWO years of school: 2 years for an Associate's in Physical Therapy
(AT) Athletic Trainers go to FOUR or SIX years of school: currently, it only requires a bachelor's in Athletic Training but anyone who starts school in 2022 or later must complete a general BS + 2 years for a Master's in Athletic Training.
All three careers have a lot in common, and you probably worked with all three at different times (which is where you might have mixed up the term physical trainer.) Generally, ATs are the people who travel with sports teams and sit court side to help immediately with injuries, PTs/PTAs are more likely to work in a clinic/hospital and treat athletes after a severe injury or orthopedic surgery.
Here's more info on each:
Let me know if I can answer any questions from those webpages!
Best of luck,