I think there is zero chance that the Supreme Court chooses to decide what "High Crimes and Misdemeanors" actually means. They will say that is up to Congress to decide if/when it chooses to impeach and try a President.
That has always been for the representatives of the people (Congress) to decide. 'High crimes' does not mean what many believe it does. They are not traditional crimes like assault or jaywalking. According to Hamilton in Federalist 65, they are violations against the public trust.
Take for example the first attempt at impeachment in our country, John Tyler. He was attempted to be impeached simply because he took different positions as Harrison, and it was believed by Congress, because he was appointed through Harrison's death, he had to follow Harrison's policies because be wasn't elected. That attempt failed but the next impeachment, Andrew Johnson was successful and very similar.
Johnson's impeachment was because he fired people in the War Department that Lincoln appointed (again, Congress thought because he wasn't elected, he had to not change Lincoln's policy, even though by the book, they were people the president could fire). Also, 'disrespecting Congress' (actual term in the articles) by basically thumbing his nose at their demand of oversight.
Historically, a president doesn't have to commit a serious crime, he just has to, in the opinion of Congress and the founding fathers, betray the trust of the people (what Hamilton called the 'high crime of the presidency').