The title depends on the faculty/department. So Dr. rer. nat. is only given in biology, physics, chemistry and mathematics. At some universities these departments give more specific titles, so for a biologist a Dr. rer. nat. is equivalent to a Dr. rer. biol.
A Dr. rer. medic or Dr. sc. me. or Dr. sc. hum. is given by various medical faculties for (natural) scientists that may not treat patients. Only if you are allowed to treat patients after the "Approbation", you can use the Dr. med. you earned at the faculty ("medical doctor").
More and more one can also get a "Ph.D. in blabla" degree certified instead of a "Dr. rer. bla".
Note that there are different customs whether one tries a Dr. E.g. in engineering, it is quite rare for an engineer to have a Dr. Ing. Only for high-rank faculty staff ("Chefingenieur", second to the professor who is head of the institute) and of course for the professors a Dr. Ing. is usual. In chemistry, however, more graduates proceed to a Dr. rer. nat.
Most of the time, only the Dr. is used, so one cannot tell if it is actually a Dr. rer. nat or Dr. sc. hum. (The only exception are the medical doctor Dr.med., Dr.med.dent., Dr.med.vet. as the indicate also the "Approbation".)
It is almost never possible in Germany to start a Ph.D. programme only as a B.Sc. graduate.
I assume a Dr.rer.physiol. would be given by a medical faculty. It may happen that with a Dr. from a medical faculty, it's more difficult to get the "Prüfungsberechtigung" (the right to give grades to students) at a natural science faculty even if the research field would match.
However, all this knowledge is quite useless for you to plan your own career. The choice of subject and supervisor is more important to you. This would automatically determine the department and thus the title you would get. If you choose both only for some letters in front of your name, your scientific interest in the subject is too low to endure all the work and frustrations. You can't plan scientific results, so you can't plan a scientific career.