Is there a way to run Safari 5 or 6 on Mavericks?
You already know the answer based on the link you provided.


  Safari is Integral part of OS X.
  
  That is way you can not run other versions on Mavericks for example.
  
  The link you provided clearly shows the compatibility.


Quote:
 Will not work, per Multi-Safari website: "Safari normally uses the Web Kit framework found inside Mac OS X to render web pages and execute javascript. This means that if you preserve an old version of Safari to run it on a newer version of Mac OS, it will use the newer Web Kit found in the system and you will get the same results as with the newer version. Thus, you would normally need a separate installation of Mac OS X for each version of Safari you want to test a website into." –  php_on_rails May 7 at 15:27

The solution is to have appropriate OS installed (you can have multiple OS installed) as explained here.