Minecraft Launcher 1.6.13 won't update itself

(I've set 1.6.19 as affected version because I have no other choice. This bug is affecting 1.6.13 only.)

I know that there's already a Minecraft Launcher 1.6.19, but my launcher won't update and is still on 1.6.13. The launcher log says:

[11:26:05 INFO]: Minecraft Launcher 1.6.13 (through bootstrap 5) started on windows...
[11:26:05 INFO]: Current time is Aug 15, 2015 11:26:05 AM
[11:26:05 INFO]: System.getProperty('os.name') == 'Windows 8.1'
[11:26:05 INFO]: System.getProperty('os.version') == '6.3'
[11:26:05 INFO]: System.getProperty('os.arch') == 'amd64'
[11:26:05 INFO]: System.getProperty('java.version') == '1.8.0_31'
[11:26:05 INFO]: System.getProperty('java.vendor') == 'Oracle Corporation'
[11:26:05 INFO]: System.getProperty('sun.arch.data.model') == '64'
[11:26:05 INFO]: proxy == DIRECT
[11:26:05 INFO]: JFX is already initialized
[11:26:05 INFO]: Refreshing local version list...
[11:26:06 INFO]: Refreshing remote version list...
[11:26:06 INFO]: Refresh complete.
[11:26:07 INFO]: Loaded 2 profile(s); selected 'Snapshot'
[11:26:07 INFO]: Refreshing auth...
[11:26:07 INFO]: Logging in with access token

Note that it's no network error, so it's no technical support issue.

The bug is also occuring on the computers of other members of the German Minecraft Wiki community (see here) and it seems that the "old" launcher which required Java already being installed manually and was installed without the install wizard only is containing this bug.

Note that I'm on Windows 10, but in the launcher log it says 8.1. This bug is falsifying the statistics!

I've searched for "1.6.13" and "update" and could not find any bug report about this bug. I'm sorry if I missed one and created another duplicate.