You mention "two headsets" and "a 1 inch threadless headset". What dimension head tube is the 'other' headset made for? And what, after "thinking" do you think the answer might be?
The link text says 1 1/4 to 1 1/8 yes I know, but if you look at the webpage the actual item in the size options list is a 1 1/8 to 1 reducer.Above item is 1 1/4" to 1 1/8" reducer - the OP has a (new to them) 1 inch steerer tube along with a 1 inch threadless headset to fit into a 1 1/8" head tube aiui.
Q&A:the actual item in the size options list is a 1 1/8 to 1 reducer.
Also check the questions and answers lower down the page