Credal orthodoxy, unbroken apostolic succession, uninterrupted true teaching about the Eucharist (yes, all three are Catholicism's criteria for valid orders) and the Eastern liturgies, Byzantine, Armenian, Assyrian, Syriac, Coptic, Ethiopian, Malankara, and Maronite. Using just the Vincentian canon (what has been believed always, everywhere, and by all), you pretty much get Catholicism; the Christian East is Catholicism in Eastern (Greek and Middle Eastern, and later Slavic) forms: very little defined doctrine but lots of beliefs and practices the same.
Communions of churches, the Orthodox and the Oriental Orthodox, respectively, remarkably little to do with each other.
Eastern Christians: Canonical Orthodox, non-canonical Orthodox such as the Kiev Patriarchate, Orthodox splinters/outliers such as the Old Believers, Copts, Armenians, Assyrians, Ukrainian Catholics, et al. You get what I mean.
Not really Eastern Christians: Western vagantes who claim to have Eastern lines of succession (an un-Eastern way of looking at holy orders) because their founder 100 years ago got an Eastern bishop to ordain him, then left that church to become independent. Some Western vagantes have taken to trying to pass themselves off as Orthodox.