Apple ID - Security questions reset

When I'm logging in my Apple ID (http://ift.tt/1mBZW12) in the tab "Password and Security" there are security questions, just like this article says: http://ift.tt/1wXN0YB

The thing is - I know answers to my security questions, I keep them in text file, just like my other accounts data (e-mails, game accounts and so on), so I had no chance to forget them. But nevertheless my answers are not right according to what the page says when I click "Continue".

Ok, I thought, then I'll try to reset my security questions. I read the article that I mentioned before (HT201485), but the thing is - there is no any link to "Send reset security info email to [rescue email address]". And I do have 2 additional e-mail addresses.

I also checked the source code of the page. And guess what I've found:

"TODO add ability to reset CRs via rescue email"? That's above good and evil.

I know, that's very likely a duplicate, but I have some information that might help others with simular situation (in the my own answer below) and also I have a question:

How is it possible that's my 101% correct answers suddenly became incorrect?