The way the rules are structured, a quarantine is mandatory for everybody but it can be “shortened” once the German authorities have received a proof that you tested negative, are fully vaccinated, or recovered from Covid 19. The rules for so-called “variant areas” and “high-incidence areas” are slightly different. In any case, there is no blanket exemption from the quarantine requirement and the German authorities want to make that determination themselves.
The official definition of “vaccinated” is based on the recommendations of the Paul-Ehrlich-Institut. You must have received all prescribed doses at least 14 days before travelling/ending your quarantine.
In practice, you still need to register electronically, go straight to your place of (temporary) residence but would presumably be allowed to go about your business as soon as you have provided a proof of vaccination. That also means you will need some sort of proof of vaccination in a language they accept (currently German, English, French, Italian, or Spanish).
Incidentally, the last regulation refers to some scientific justification for the rules. They suggests that vaccinated people are typically contagious for a shorter time if they get infected (rather than altogether non-contagious), hence a “reduced quarantine” even if it can feel like an exemption in all but name.