Well, I wouldn’t say " nobody is certain exactly which one is really the unbroken chain because it happened so long ago and is lost in history". The Orthodox are sure that the Orthodox Church is the one true Church, and the Catholics are sure that the Catholic Church is the one true Church! You’re right about the “split hundreds of years ago” part. The conventional date given for the East-West Schism (or “Great Schism”, as it’s often called, though I fail to see what’s so great about it :p) is 1054 AD, but that’s really a matter of convenience, because that was the year when legates from Rome, who were representing a Pope who had actually died while they were on their journey to Constantinople (so some people say they had no authority), placed a “bull of excommunication” on the altar of the Hagia Sophia basilica in Constantinople, excommunicating all of the Eastern (Orthodox) churches over various issues that the Latin and the Byzantine churches had grown estranged over. The Byzantines responded in kind, excommunicating the Latin Church. These mutual excommunications were lifted in 1965 under Pope Paul VI and Patriarch Athenagoras I of Constantinople, who met in Jerusalem in 1964 to discuss matters and eventually issued the “Catholic-Orthodox Joint Declaration of 1965”, which rescinded the excommunications of 1054. This did not end the Schism however, as it was about much more than that initial act by the Pope’s legates (indeed, the issues surrounding the division, such as the Filioque or the place of Constantinople in the Pentarchy, had been simmering for centuries before 1054).
That, in a nutshell, is how “Roman Catholicism” and “Eastern Orthodoxy” came to be distinct, separate churches and communions.