A seatpost 0.5mm undersized will definitely be far too loose when inserted. But I don't think that's the issue here.

A 13/16" seat post should be 0.8125" or 20.6375 mm in diameter. If it's measuring 20.72mm, then it's around 0.1mm oversized - not undersized - IF that measurement is accurate.

However, you also might want to check your calipers. A 7/8" seatpost should be 0.875" in diameter, or 22.225mm (22.22mm to the limits of your caliper's scale). Instead, you're showing it as measuring 22.32mm - or about the same amount oversized as the other seatpost. To me, that suggests the possibility of a systemic measurement bias of approx +0.1mm.

FWIW: I find that my digital calipers often tend to show off by about 0.1mm (e.g., register -0.1mm or 0.1mm when fully closed) if I don't zero them before taking a measurement.