Hi Thomas,
What we call the ‘main menu’ is the black menu along the left, where you can go to Open > Time Dimension. The beginning of that article shows creating a new time dimension from the main menu clicking New instead of Open. In the Explore window, you can also right-click a time dimension found under the “Time Dimensions” folder to edit it. Then you can set Hierarchy Level to Day to customize the format for the Day level for when you have a table like the one shown in your first post.
This is for customizing the formatting of a time dimension hierarchy such as Year > Month > Day, but your screenshot shows the dialog accessed from the Data Analysis Panel for customizing the Day Of Week attribute, which is sort of the alternative you found later to instead drag each component of the date separately (Month of Year, Day of Month, etc.)