How To Force Iphone Backup

I have an iPod5. I have been using iCloud for backups. However, I have an issue in a particular app that only goes away when I fully reset the device. (The issue in the app persists if I restore from iCloud backup, but that's not what I want addressed here). So, at the moment, my iOS device is running as if it were a fresh new iOS install. I want to be able to keep going with this install and have it backed up to iCloud. However, I also want to keep my old backup around so that I can go back to that later.

I thought that iCloud would automatically start a new backup series when I chose to set up my iPod5 as a "New" device. However, when I look at the backups UI, I get the impression that I am going to lose my old backup as soon as it performs the next backup.

Under iCloud → Backup, I see "Last Backup: 1:37 AM". That is the timestamp of the backup I performed on this iPod before resetting it:

Backup / Back Up Now / Last Backup: 1:37 AM

Under "Manage Storage" when viewing iCloud settings through the device, it lists the iPod I'm using as matching the backup I made before I set up my iPod as a new iPod. The backup that it says is associated with this iPod is the very one I want to preserve when I make a backup of my iPod:

Manage Storage / Binki's iPod Touch / This iPod

And when visiting that screen, you can see the discrepancy between the backup that looks like it will be overwritten and the backup that iOS plans on overwriting it with:

Info / Latest Backup: 1:37 AM / Backup Size: 715 MB / Next Backup Size: 16.0 MB

So, how do I get iOS iCloud backups to start a new backup or save/checkpoint/rename the old one? I'm starting to fear that my only option is to restore the old backup, make a backup in iTunes, and then manage storage of that backup manually which is a pain that I hoped iCloud would be able to prevent…

In an answer to another question, I read: