MacOS – OS X 10.9.4 shows wrong used disk space

Most of the difference you're seeing is due to Time Machine's "Local Snapshots" feature. When Time Machine is turned on but the backup device isn't available, it backs up to the local volume. The space used for these local snapshots is counted as "in use" by Disk Utility and System Information, but not the Finder (see the Disk Space considerations section of this article). While the space for backup actually is in use, it'll be freed automatically when needed (i.e. when the volume gets above 80% full), so the Finder counts it as being available.

In your particular case, the space System Information lists as being used for "Backups", 98.36GB, is very close to the difference in free space listed by the Finder vs. the other two. The Finder's 206.43GB free - 98.36GB of backups = 108.07GB actually free; compare to System Info's listing of 108.44GB free and Disk Util's 108.31GB. I'm not sure what the rest of the difference is (maybe they looked at the disk at slightly different times? Or they may be counting volume structures a bit differently?), but it's very small.

Finder shows:

• easily, the truth for JHFS+ volume Macintosh HD
• optionally, the truth for mtmfs volume MobileBackups

Hint: with local snapshots enabled in Time Machine, in Finder you can go to /Volumes/MobileBackups then get info.

Results of a one-line command show that for some purposes, MobileBackups is treated as a distant file system:

qlmanage -m disks | grep MobileBackups && mount | grep MobileBackups

As Disk Utility is oriented to local file systems, we'll probably never see MobileBackups as a separate volume in that context. There are degrees of simplification in Disk Utility, even when debug options are chosen.

Ok, so after posting this question I had some major breakthroughs, most of which are posted in the update to my question. By then remembering that du still reports in units of 1024 whilst OS X now uses 1000, the final 7 GB was accounted for (well, there's still a potential 500 MB missing judging by Disk Utility, but that's probably just an artefact of du counting in blocks). Now the only question left is why System Information is reporting a substantially lower value for Backups than it should be, and whether or not it will remember to eventually clean up the other 18 GB of local snapshots.