It looks like the project from which you made your screen capture is hosted on BIM 360, not Autodesk Construction Cloud [ACC], which is what is shown in the screen capture in the Help article you referenced.
In my firm's projects, I also do not have the more button for projects hosted on BIM 360. We do have at least one BIM 360 project that is authored in Revit 2023. I do have the More button (three-vertical dot icon) in projects hosted on ACC, and for those in Revit 2023 or later, I can start the data exchange process.
I am not 100% certain, but I guess that the data exchange feature is not available on BIM 360, only ACC.
David Koch
AutoCAD Architecture and Revit User
Blog | LinkedIn
