> it will happen to coincide with the birth of ... Jesus Christ.
More precisely, it will happen to coincide with what somebody incorrectly calculated to be the birth of Jesus Christ, but which we now know wasn't.
Which is now calculated to be between 6 BC and 4 BC.
That seems to be the current consensus, yes, although I don't think later years can be conclusively ruled out.
It became popular among some English protestants who didn't like to use Anno Domini because it's Latin.
I think it's also popular among some Jewish people, because they're OK with giving dates in terms of Jesus but they don't believe he was Christ/the Messiah.
It is apparently fairly common to use by Jewish scholars too, though I can’t couch for that.