But isn't this different because it is a trial of impeachment? He's already been impeached.

Edit: I know this is long after the discussion, but I want to add some thoughts in response to your points.

I read the Nixon v. US case (1993) which upheld that the Senate has sole authority to carry out impeachment trials. However, I believe that's not the issue here. The Constitution makes clear that impeachment is the process to remove an officer of the US from office and that the Chief Justice must preside over the impeachment trial of a president. Roberts is not presiding because Trump is no longer a sitting president. "President" is the position that made him an officer of the US, so he is no longer an officer of the US, either.

The Constitutional question would be whether a former officer of the US can be impeached. It seems that the vote on whether it was Constitutional to hold the trial was not the Senators' call to make. The Supreme Court could rule that it IS the Senate's call, but I believe they would be contorting themselves to make this ruling. After all, once someone is no longer an officer of the US, he/she returns to being a private citizen. There are other legal actions that can be taken against a citizen, reserving impeachment for officers, which is the prima facie Constitutionally prescribed scope of impeachment. Besides this, there's the question of how long after holding office does a person still come under the impeachment jurisdiction of the Senate.

But isn't this different because it is a trial of impeachment? He's already been impeached.