PNGs are almost always smaller than GIFs for that sort of image (flags, icons, QR codes and such), especially at smaller size. But you’re correct, I did try GIFs too - you never know, they might be smaller in some circumstances 
There are smaller file sizes with different formats, but they’re all lossy compression schemes that are totally unsuitable for simple geometry shapes in an indexed color space such as a flag design or a QR code. But for images that involve mostly human interpretation of fuzzy features, such as a photograph to put in a vCard on your tag for instance, WEBP works wonders. See this:
The compressed Amal is 555 bytes!