Update: I just ran a backup on my MacBook to verify this.
- Empty space went down from 113.6 GB to 82.1 GB, a difference of 31.5 GB.
- Space taken up by Applications went up from 86.2 GB to 117.7 GB, a difference of 31.5 GB
- Space taken up by iOS files went from zero to 31.5 GB
Looks like this confirms the accounting error. The backup files are only stored on your drive once, but count towards both Applications and iOS files.