How to change the home dock icons on TouchWiz? Similar to How to edit the home dock icons (shortcuts) on TouchWiz 4.0 ICS

I have a Samsung Galaxy S3 GT-I9300 with Android 4.0.4. I can't for the life of me figure out how to edit the home screen dock icons on the default launcher (Touchwiz).

I've tried:


Long pressing on a home dock icon (nothing happens).
Long pressing an app in the home screen to drag onto the home dock. (Home dock icons disappear and are replaced by a Delete bar across the bottom of the screen)
On the home screen, press Menu -> Edit. (Home dock icons disappear and are replaced by 4 icons: Create folder, Create page, App info, Delete).
In the Apps list with a Customizable grid view, press Menu -> Edit. This is the answer to the linked question. (Allows editing of the Apps listing, but doesn't bring the home dock icons onto the screen so I cannot modify them)


Before I updated to 4.0.4 I could change the shortcuts in the home screen dock, and I put Google Chrome there to replace the default browser. After updating I now want to replace it with Firefox.

I am in Australia. I have friends with grey imports of the same phone from Hong Kong that don't exhibit this behaviour on 4.0.4.
The way it works on my SGS2 Android 4.0 is


Click on applications,
Click menu button, select "edit"


Now the screen shows several app icons you can change, but at the bottom are also the home buttons. All home buttons except for the most right one ("start page" or "home page") can be dragged and dropped, and you can move other app icons to the row with the home buttons.

Note that this only works if at step 1 you are in grid view, otherwise change to grid view in the menu first.

If this doesn't work, there may be some additional carrier software that prevents it.