Mostly agree ... just being nitpicky here
The AMA is a professional organization. For MD schools in US and Canada, it's the Liaison Committee on Medical Education (LCME) that accredits MD schools and they are in no way affliated with the AMA. The US Dept of Education recongize LCME as the only body to accredit MD schools, just as the US DoE recongize the AOA as the only body to accredit DO schools. (the confusion lies because the AOA serves as the accreditation body as well as a professional organization. The AMA is only a professional organization)
Yes, your degree is solely based on the degree granted by the institution that grants your degree. If your school granted a Bachelor of Arts degree in Chemistry, then your degree is a BA (or AB), even if you took more science courses than someone with a BS (or SB). If MIT awarded you a Doctor of Science degree in Chemistry instead of a PhD in Chemistry (even though the requirements are exactly the same), you have a DSc. (or ScD), and not a PhD.
In dentistry, if the school grants a Doctor of Dental Surgery, then the degree is a DDS, even though the coursework and requirements are exactly the same as a DMD (Doctor of Dental Medicine). For dentistry, it's purely based on the school whether it gives a DDS or DMD.
Foreign doctors who have not earned the Doctor of Medicine degree are not granted an MD at any point while the US. Basically what happens is that the state boards and various organizations have looked the other way when it comes to them using "MD" for patient care. So someone with a MBBS (or MBBCh) can use an MD and people will look the other way. In academia, people care and the same MBBS (or MBBCh) cannot use the MD title in academia ... whether it's a Journal Publication, name tag, CV, etc. People in academia are quite stingy when it comes to correct postnominal letters.
Bascially if the state board of medicine, state board of osteopathic medicine, the AOA, the state chapter of the AOA, and local medical societies started not to care if DOs start using MD, then you will have the same situation of DOs using MD title that you see with MBBS people using MDs. But it is highly unlikely in the near future given the resistants by the old guards and even some of the younger members of the profession.