I downloaded the .ipsw for my phone (5s gsm) from this site: https://ipsw.me/ (you select device, then the iOS version; you want one of the green ones, they are still signed by Apple). I used my computer (an iMac running OS X 10.6.8) to download the ipsw, and placed it into a folder (you can put it on the desktop as well).
I then CLOSELY followed the process at http://www.macworld.co.uk/how-to/iosapps/how-downgrade-from-ios-9-to-8-7-reinsta ll-ios-8-3522302/#howtodowngrade
...particularly the "hold down the Option key and click on the Restore button" (took me three tries: first two times I just clicked on Restore and it tried to give me iOS 9 instead of letting me select the .ipsw i downloaded...but with Option/Restore, it let me pick the 8.4.1 -- iPhone6,1_8.4.1_12H321_Restore.ipsw)
After it restored the phone to 8.4.1, iTunes again recognized it, and asked if I wanted to restore from the backup (YES, YES, PLEASE ) ... and everything is now back to normal.
p.s.: yes, the same RESTORE verb, but different context--
--the first RESTORE, Restore iPhone, changes the firmware, is the one that replaces iOS 9 with iOS 8.4.1, leaving you with a totally wiped phone, just like when it came from the store
--the next RESTORE, Restore Backup, grabs all the data you saved the last time you sync'd (I automatically sync every time I plug the phone into the usb port to charge) and puts it back onto your phone