Following your help Lawrence, then with subsequent guidance from Applecare, I've successfully restored the iPhone from the Time Machine backup. The first Time Machine backup I selected failed to copy - I got a heart-stopping error message half way through. But for some reason it worked with a different backup which had been created earlier on the day before the iTunes backup had been overwritten.
Many thanks for your guidance with it, Lawrence.
So what have I learned from this? If I had not had a Time Machine backup, I'd have lost all the data. And that's because when Applecare advised me to factory reset the iPhone to try to diagnose why the iPhone was often not receiving calls, the advisor omitted to warn me to check the box in iTunes to prevent automatic backups when the phone is connected to the Mac. (Next time I connected it, it overwrote the single backup - Apple's guy confirmed today that there is only one, which is updated by each new backup) .
So unless I've got some of this wrong, if you fail to check that box in iTunes and then connect a wiped phone (perhaps as part of a similar diagnostic mission) you lose all your data. Surely there should be far more built-in protection against such a potentially catastrophic data loss?
Thanks Lawrence, you saved my life today !
I had a corrupted backup file on itunes, and I reinitialized my iPhone by error.... So everything was damaged and I could only start my phone as "new" = empty.
By following your advice I managed to make it work.
One important thing on itunes ( Mavericks) that you don't mention : before connecting the iphone I must deactivate the automatic sync in itunes pref, otherwise, my Time Machine recovered file is automatically replaced by the empty iphone content ...