Administrator user can't visit pages in Safari as they are not approved in Parental Controls
The problem was related to the fact that Parental Controls were enabled for my account (A) under Users &amp; Groups but I was unable to disable it since the checkbox was grey. Also, the Parental Controls in System preferences didn't shown that anything is set.
Since the only user that can disable the checkbox is another admin I did the following:


Created a new admin account, lets call it B,
Logged in with the account B,
Disabled the option for the account A,
Logged out from the account B,
Deleted the account B.


Once I did that, everything that did not work started to work without any issues.

The affected stuff were:


Web pages I could not visit,
Siri deactivated w/o possibility to be activated
Dictation deactivated w/o possibility to be activated
Captive unable to load the login page on WiFi hotspots