Niels, those are Reader Spreads, which do not represent how a printer would actually print a booklet.
Nicole, I don’t know anyone who uses the InDesign print booklet feature. As you can see, it doesn’t work very well. If it’s a small piece, I just export the PDF and place in an InDesign spread like this: https://i.imgur.com/Cu6L5CO.jpg (If you have bleeds, you need to pull in the insides, since there are no bleeds on the inside of a booklet.)
Basically, imposing from a PDF is the way to go. If you just want to print it, Adobe Reader/Acrobat has a booklet option.
There are many PDF imposition plug-ins out there (some for InDesign), maybe someone here can recommend. (Sorry I haven’t used any of these recently to know)
From Adobe:
One such set of Acrobat plug-in products are Quite Imposing and Quite Imposting Plus (available for both MacOS and Windows) from Quite Software https://www.quite.com. Another similar product is PDF Snake (Windows only) available from https://www.pdfsnake.com.
Otherwise, when you are searching, “imposition” is the word you should use.