Depending upon what country that you are in then you might be able to re-download music purchases on your other computer, as described on the page that you linked from (music can't be re-downloaded in all countries). You might also be able to turn on 'automatic downloads' so that any future purchases are downloaded automatically on that computer : automatic downloads.
Audiobooks (and ringtones) and currently one-time only downloads from the store - I believe that they are all supplied to Apple from audible.com, so I assume that it's them who are requiring the one-time download. To get it on your other computer you will need to copy it to it yourself e.g. via home sharing, external drive, flash drive etc : http://support.apple.com/kb/HT4527