Both variants are fully authentic. Riveted chain mail takes more material and time to produce, so was more expensive. However, it works a lot better at stopping thrusting attacks, meaning mostly arrows and bolts. As this was the main threat on many battlefields, war chain mail was mostly riveted. When expecting to fight mostly bladed enemies, such as in tourney or parade settings, butted mail was not uncommon. And once guns started to appear on the battlefield, it quickly became apparent that chain mail would not stop bullets, so the butted style got more popular again, as it served to still protect in melee combat.

As for pinching: chain mail was never supposed to be worn on bare skin anyway. At least a tunic, but usually heavier, densely padded wool clothes ( often called gambesons ) were worn underneath. This not only prevents any chafing or pinching, but more importantly it helped soften the impact of any blow a bit.