For the EU certificate, proof of prior infection can be used, although I believe you need a positive PCR test for that, an antibody test is not sufficient.
> ... although I believe you need a positive PCR test for that, an antibody test is not sufficient.
Most countries in the EU just relaxed the requirements to travel before the summer vacations. You don't need a PCR test anymore to travel: just an antibody test (at least for Spain / France / Belgium but I take it many others are doing the same). And there's no requirement to be vaccinated.
They also raised the age at which you need a test from 6 to 12 years old.
Source: always living/traveling across France/Belgium/Spain.
Fun fact: when you travel by car in the EU they typically do not bother to check. You can cross several countries and they're not verifying anything. Another fun thing: test is supposed to be less than 72 hours old, so if you're traveling over more than 3 days, you need to get re-tested during your trip.
Antigen tests (the rapid test kit, results in ~15 minutes, sample taken from the nose or thread) are not the same as antibody tests (blood test for prior infection, has to go to the lab). You need the former for travel. Afaik, the latter is not accepted as proof of prior infection, you need a positive PCR test from at least 28-ish days ago.
The DGC doesn't have any such type of certificate, so there's that (which, obviously, doesn't mean countries cannot accept antibody tests). As for the traveling, that's weird, as I thought most countries exempt you from needing a test if you're just transiting.