If you have already received the 1st vaccination, you are exempt from the obligation to test for a maximum of 3 months from the 22nd day after this. Fully vaccinated people are exempt from testing for 9 months.
Vaccination certificates, a vaccination card, "greencheck"-app or printout are valid as proof of vaccination.
Vaccines for which only one vaccination is intended (e.g. from Johnson & Johnson), the exemption from the obligation to test applies from the 22nd day after the vaccination for a total of 9 months.
Please note: All vaccinations of vaccines that have been approved by the EMA or have successfully passed the corresponding process of the WHO are recognized as proof of vaccination. Currently (June 10, 2021) these are the vaccines from Biontech / Pfizer, AstraZeneca, Covishield, Johnson & Johnson, Moderna and Sinopharm.