Thanks for your reply, it was really helpful and got me looking in the right direction.
Investigating a little more: Like you said, the Redo only shows the Data Table Window option if there are multiple reports, and it will show private tables.
The status bar icon doesn't show the button when you have a private table, so works the same as the shortcut Ctrl+Alt+D. You can also find this option under View -> Associated Data.
The most consistent way I found is to use Window -> New Data View. (Just a note: This button makes the table visible if it's private or invisible, but will create a new table/view if the table is already visible.)