The Framers didn’t want the impeachment power to become a political weapon. That’s why they designed both procedural and substantive protections against misuse of this important legislative check on the executive. The procedural protection is the requirement of a two-thirds vote for removal, which makes it impossible to remove the president without broad support. The substantive check is the list of offenses justifying impeachment.
The words “other high crimes and misdemeanors” does accord Congress some discretion, but not as much as the rejected term “maladministration” would have. The words would seem to require criminal-like acts of a serious nature, though precisely what would suffice is anything but clear. A sitting president would almost certainly be impeached if he committed murder, despite the historical precedent that Vice President Aaron Burr was not impeached for killing Hamilton in a duel. But if a president paid hush money out of personal funds to prevent his adultery from being disclosed—as Hamilton did when he was Treasury secretary—he wouldn’t be impeached. Adultery was a felony in Hamilton’s time, but nothing Hamilton did constituted a public crime. Perjury to cover up adultery—one of the offenses for which Mr. Clinton was impeached—is a closer call, although I believe it was not impeachable.
As for the allegations against President Trump, obstruction of justice is plainly a high crime, but a president cannot commit it by exercising his constitutional authority to fire or pardon, regardless of his motive. (It would have been an impeachable offense in Mr. Clinton’s case, but the facts were disputed.) Neither is it a crime to conduct foreign policy for partisan or personal advantage—a common political sin with no limiting principle capable of being applied in a neutral manner.