I have an old iPod Touch with old software that can no longer be updated:
iOS version 6.1.6
I've been searching for a solution to my iMessage login problem for months and tried everything from those frustrating links and all the tips from other users.
Here is what finally worked:
1) Settings > Privacy > Location Services > On
2) Settings > General > Date & Time > Set Automatically > On
3) Reboot device until Apple logo appears
4) Settings > Messages > Sign In