I have seen references to all three of these applications on various parts of the Qt website but am completely unclear as to the exact differences between them and whether they are actually separate things or just different names for the same thing, or the name changed over time? Or is one no longer supported? What's the deal with these?
Qt Creator is Qt's IDE. You don't have to use it, but it greatly simplifies Qt development.
Qt Designer is a graphical tool that lets you build QWidget GUIs. Qt Quick Designer is similar, but for building QML GUIs. Both are built into Qt Creator.
This is explained in a little more detail over at Wikipedia.