Which generation is my iPod touch?
You do have a 5th Generation iPod touch, and it's the 16GB model.

If you check out Apple's website for current-generation iPod touches, specifically the "tech specs" page, you'll see that there are two types available today. The left one (the one with no metal circle on the bottom-left of the back) is the one you have.

On your Usage page, you have two numbers: available and used. The confusion is probably because "available" here means how much space you have left for new content, not the total amount available to you. If you add the two numbers together, you'll have roughly ~13GB of total storage. (the remaining storage being taken up by the OS)

Also, Apple has an article on identifying your iPod that can often help find the marketing name for any iPod that's been released.