That looks like you have a separate partition named "MacBook Air (internal)".
Start up the Disk Utility app (in your Applications/Utilities folder), and see what it shows in the sidebar.
Each physical drive is shown, with the partition(s) on it indented underneath it. In the sample, I have a 640 GB internal HD with 2 partitions, and a 320 GB external with 6.
Clicking on an item in the sidebar will show various info about the disk or partition.
I think you'll find the MacBook Air (internal) is a partition on your SSD. If so, click it and see how large it is, how much is on it, etc.
That partition may also show on your destop or Finder sidebar. If so, open it. I suspect you'll find another Backups.backupdb folder there.
EDIT: Sorry, I hadn't seen the inset at the bottom of your attachment clearly:
It does look like MacBook Air (internal) is the only partition on your SSD (they're usually named Macintosh HD, even though they aren't really HDs).
Since you've deleted Backups.backupdb and done the full reset (you did quit System Prefs while you did that, right?), I don't know why it's doing that.
Two questions:
When you disconnect the external, are you ejecting it first? That shouldn't matter, but you never know.
Are you running any 3rd-party apps that deal with Time Machine, or using something besides a "normal" external HD (like a NAS device)?