I'm wondering if anyone would care to share what these two terms, the Christian East and Eastern Christianity, mean to you? (Also, do the two terms mean the same thing?)
For me those are basically historical and geographical terms. It means EOs, OOs, Eastern Catholics and Nestorians.
What I get from Peter J's quotes is that it is a vague term which can mean different things in different contexts, such that the original general question is largely unanswerable.
In some contexts (actual geography, historical circumstances of dealing with Muslim conquest) the ACoE are Eastern Christians, while I (American convert) am not. But in other contexts (when 'Eastern Christian' is being used to refer to the theological tradition running from St. Athanasius through the Cappadocian Fathers to St. Maximus and St. John of Damascus down to St. Gregory Palamas--i.e., as primarily expressed in Greek and contrasted with the developments in the West), I am an 'Eastern Christian' while they are not.
In the same way, there are certain senses where a 'cradle' Byzantine Catholic born and raised in Pittsburgh is definitely an 'Eastern Christian'; certain senses in which he is definitely not, and certain senses in which whether he is or is not is disputed (i.e., when 'Eastern Christian' is being used as a historically understandable but theologically weak synonym for "Orthodox).
As such, when a question such as "should I be an Eastern Christian" is asked without context, it's impossible to answer.
"Do you mean a Christian living in Eastern Europe or the Middle East?" Answer: I don't know, do you *want* to live somewhere in the east? (nothing to do with religion)
"Do you mean a Christian with family and cultural ties to the Eastern Europe or the Middle East?" Answer: unless you are born that way, you can't be. don't be silly.
"Do you mean an Orthodox Christian". Orthodox Answer: Yes. Latin/Byzantine Catholic answer: No or "you can be without actually joining an Orthodox Church"
"Do you mean a Christian that uses a liturgy historically associated with the East?" Orthodox answer: if you aren't going to adopt the theology that goes with it, then it's irrelevent.