But from what I've read, it's always been clear, including for the Founding Fathers, that it wasn't required for the offending actions to be criminal. And since just about everything about impeachment seems to be defined by the historical precedent, there have been articles of impeachment that didn't even remotely involve criminal actions.

As I wrote, impeachment is a political judgement call, so certainly there may be politicians who set the bar higher than others, requiring actual crimes and basing their judgement on the 'innocent until proven guilty' standard. Fair enough if so, though it does rather seem like both parties in the two recent impeachments have been guided more by partisanship than by such theories.