This site is correct. Natacha is a pet form of the Russian name Natalia. (In both Natacha and Natalia, the second "a" is stressed.) A name can have several pet forms, however.
Russians often use foreign forms of the same name as pet forms. E.e. "Zhora" is a pet form of George, probably from the French form of the same name.
If we consider the name Natalie with its English (the stress on the first "a") or French (the stress on "i") pronunciation, Russians feel those forms as foreign. Thus, Natalie (either English or French) can be used as a pet form of the Russian name Natalia (or Natalya, which is the same).