Re: Rename the terms?
"High crimes and misdemeanors" means whatever the sitting Congress wants it to mean; the terms, as they relate to impeachment, are not defined in the US Constitution.
In theory, Congress could impeach the president 'because he looks funny,' though that would be a patently ridiculous thing to so, so it hasn't happened thus far-- we'll see if that changes after the next Democrat takes office.
With regards to the difference between misdemeanors and felonies in criminal law, their distinction is also up to the legislature (Federal, State, and local, and thus vary slightly between locales), but they generally refer to minor (public intoxication, disturbing the peace, and licencing violations) and serious (battery with serious injury, robbery or fraud in the thousands or more dollars, and murder) offences, respectively.
It's entirely possible, however, for the same crime to be a misdemeanor in one location but a felony in another. It's even possible to have a crime be both in one location, with the choice being up to the prosecutor. (These are called "wobblers," if you want to look them up.)