What are the proper input notations for the Go To XY tool for each of the tools drop down options?
When set to the the Decimal Degrees it's fairly simple.
but when set to UTM
despite the variations I've tried it has remained greyed out.
Some description is provided here but admittedly not specifically UTM: Go To XY—ArcGIS Pro | ArcGIS Desktop
But one trick to help answer your question about UTM format is given here "When a new unit format is chosen, coordinates in the recent history list are converted to match the new format."
So if I've found a location in DD, when I switch to UTM, I get my previous entry converted:
Another trick in terms of understanding format is to right-click on the map (with the Explore tool active) > Convert Coordinates > UTM
You'll get something like this: 31N 467138 0049824
Retrieving data ...