Jan and Antoine, that is helpful information (I hope?). Here is the status of my problem solving... I worked backwards and deleted the common apps that Gozen and I shared (as I listed previously), rebooted the iphone, and FT still did not work. I have not done a complete restore as I am hoping we can find the culprit. I think Antoine is right - that it is a setting. Speciifically, I am wondering about our passwords (and/or passcodes), here's why:
In an attempt to further examine the software update, I used my mom's iPhone 4 (Verizon is her provider, so no SIM card). She was using ios6.1.2 and FT worked. I upgraded her to 6.1.3 and FT still worked. However, before I upgraded, I clicked on "learn more" and this update corrected two things - and the first thing may be notable: 1) "fixes a bug that could allow someone to bypass the passcode and access the phone app", 2) "improvements to Maps in Japan". I wonder whether there may be something related to our passwords that is causing FT to stop working, particularly since Jan had limited success with changing hers on her iPod. (BTW, Jan, is your niece able to FT with her phone to other people?)
When I went to change my Apple ID password, I learned there are now some requirements for choosing a password and, incidentially, my password does not fit them. Here they are (it's a laundry list):
1. have at least one letter
2. have at least one capital letter
3. have at least one number
4. not contain multiple identical consecutive characters
5. not be the same as the account name
6. be at least 8 characters
7. not be a common password
8. not be used in the past year
So, I think the next step to test the hypothesis that it's password related? The process may look something like this (though I have never done a restore, so Gail/others, please confirm proper process? - change AppleID password to meet requirements, backup everything to icloud, do a factory restore, check to see that FT works, with "blank" phone, then sync with icloud to restore everything - including settings/new password. Then try FT to see if it works. (If not, then keep checking for other posisble "settings" issues?)
This may be my next step, though it's a bit too early where I live for me to FT anyone so it will have to wait until later. But for now, I think we may be getting closer to - something? I find it curious that there are new password requirements for my AppleID and yet I have never been prompted to change it. And I'm hoping it's not coincidental tha tthe latest software update addressed a phone app/passcode issue (I know FT is a different app, and that a passcode and password are different, but it seems related? Maybe also need to consider changing phone passcode, hmmm?) Onward!