I have heard it as “Ich bin Christ”, is that right? How does one differentiate between that and “I am Christ”?
I think that is correct. Jesus Christ, in German, is Jesus Christus.
ETA: I am not German. I learned the language in high school and college, and that was a very long time ago.
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Ich bin (ein) Christ. (I am a Christian)
Ich bin Christus. (I am Christ)
Correct, yes. Personally, I would say the first sentence without the article “ein”, but that may be a regional thing (I’m from the southern part of Germany).
There probably also is a theological distinction between “Ich bin ein Christ” and “Ich bin Christ”, but that is too deep for me. 
And then there’s always the rather mundane issue of people being named Christian, who would say “Ich bin Christian”, to complicate things further.
I’d say that “Ich bin Christ.” is the neutral version and “Ich bin ein Christ.” is slightly emphatic (cf. Kennedy and his donut.)
Compare Matthew 24:5
(Luther Bible 1912)
(Schlachter Bible 2000)
(King James Version)
(American Standard Version 1901)
Wouldn’t they say ‘Ich heiße Christian’?
There’s a slight difference:
“Ich heiße Christian” = “My name is Christian”
“Ich bin Christian” = “I am Christian”
You could go either way when introducing yourself, I would guess. Of course EinsteinsHund’s translations are correct, but in everyday usage the two wordings can be used interchangeably.
In high school we were taught not to say Ich bin [Name]. There’s a difference between school and real life!
Interesting. I would agree that “Ich bin” is probably more informal, and may be regarded as less polite, than “Ich heiße”; intuitively, I would rather use it with first names than family names. But it’s definitely used in everyday speech. Seems to be one of the cases where people who learn a language as a foreign language are taught rules which native speakers routinely disobey. My favourite example for that in English is the use of “would” in the “If…” clause of a conditional sentence, which to my knowledge is considered ungrammatical but certainly widespread.
And there’s another expression which I prefer when introducing myself in a formal situation: “Mein Name ist EinsteinsHund”.
Then of course there’s the colloquial form which is ungrammatical, but often used: “Ich bin der Christian”. The rule is that you don’t use the article with proper names, but it’s regularly ignored in everyday speech.
Man, German mental hospitals must be full of sane non-Germans. How do you say “I am a Bonapartist”? 
Schnitte, perhaps I’m just being thick, but I can’t figure out what you mean by the would/if thing. Could you please give an example?
Brazilians do this in Portuguese, and I still haven’t figured out if it’s proper language. They might say “I heard from the Daniel today” or “I visited the Sara.” It really sounds odd to me, and I never risk speaking like this because I am afraid there are subtle rules to it that I will never be privy to.
Isn’t Christ also a man’s name in Germany?
Actually yes. But very, very unusual. I had to look it up.
Originally, saying:
instead of (correctly)
would have been a speech pattern employed by young, cool, free-spirited individuals, maybe hippies. The kind of people who use the word dude a lot (in American English). Today, it has become kind of ubiquitous, but it still is considered informal.
My impression is rather that it varies by region. In the south, such as Bavaria, it’s quite common even among old people to use the article with proper names, whereas in the north it’s frowned upon. I vaguely recall reading somewhere that the line that divides regions where this is common from those where it isn’t is one of the usual isoglosses used in German linguistics to differentiate southern from northern dialects, but I couldn’t google it up right now.