Not sure what the dispute is, but the case of Walter L. Nixon v United States 1993 is pretty clear that the Senate can use whatever rules they want and normal judicial procedures are irrelevant.
Just so, but the implication was that a trial in the Senate was a free-for-all with no leadership and that is not the case.
When I looked up your reference to Judge Nixon, it suggested looking at Powell vs. McCormack 1969 and that really made me feel old to remember at a glance that was Adam Clayton Powell and Speaker McCormack. So long ago now. Fifty years is a long time.