I'm not going to make a recommendation about putting "Dr." on your AMCAS application, but like others have said once you start med school everyone is considered to be "student doctor John or Jane Doe" until you graduate and can proudly hold your medical degree above your head.
As an example, I once wore a long white coat on one of my IM elective rotations because my short coat got ripped and I had to throw it away. The long coat was the only thing I could find on short notice, and I immediately caught flack for it --- a couple residents casually but firmly said that only residents and attendings are entitled to wear long white coats while on duty. I got the message loud and clear and express ordered a new short coat so everyone would know I was a lowly med student.
So yeah, the medical hierarchy has arcane rules and customs and calling yourself "Doctor" because you have a well-earned Ph.D. is gonna get you dirty looks amongst your peers and higher ups. There was a licensed podiatrist in my class and some students were calling him "doctor" the first few weeks and he put an end to it lickety-split.