You are correct, Israel did capture the Golan Heights in 1967 when it defeated the combined armies of Egypt, Syria, Jordan and Iraq.
Your are only technically correct in stating that Syria didn't initiate the Six-Day War. Eqypt initiated the 1967 war with its full-scale movements and preparations in the Sinai -- but Egypt and Syria shared a mutual defense treaty signed just the year before (Strike One!). Prior to hostilities, the Syrian Army ordered UN observers out of the Golan frontier in preparation to honor its treaty with Egypt (Strike Two!). When Syria began massing its armor and intensified its shelling of Israeli towns, what did it expect? (Strike Three!) Syria got its ass kicked, again, and lost the Golani high ground overlooking northern Israel.
1947, 1956, 1967, 1973, 1982... It's like Ground Hog Day. Arabs refuse to give up their dream of destroying Israel, get their asses kicked again and again, and then whine about losing territory in the process. When Syrians look up at the Israeli Golan, they should remember why they lost it. It is past time for Arabs to fix their own countries and leave Israel alone.