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...