Well in either case, I'll just repeat: force push should always be avoided. Regular push without forcing can handle both of those scenarios, publishing commits to your team, and also backing up data to a safe location.
That said, making backup data in a safe location isn't what git was really made for. If you really don't want history to be there, you can use cp or rsync, or you can git clone -depth 0 from the backup machine, or just use backup software. There's "bup" which is a backup tool based on git...
That is my problem right here.
There is no way to tell the difference between "I push to make something public" and "I push to back up my data in some safe location".