OK, we are now left with the conundrum that calibre is seeing the column, there is data in the column, the sony driver is seeing the column, but the data is not being used to build a collection. I cannot say why this is happening.
One last check to make before we start doing difficult things. Please resend a book (or two) that has something in the disporder column to your 650, and see if the collection appears for those books. Even better would be to delete the books first, disconnect the reader, check that the books are really gone, then reconnect and send.
Another possibility is that something has gone wrong with the metadata.calibre file, and it is not being updated. This can be tested by deleting the file, as long as you are using automatic management. Lets look at that after send books test.
And on my system, the tag browser is on the left.
