You're broadly correct...
But i'd go as far to say that Mark was the only one on the spot and calling Luke a 'doctor' oversimplifies the greek meanings of the word.

Also that they 'wanted another one' implies some sort of massively formaized church structure rather than a small 'Johnnine' community that probably produced the Gosepel. The earliest collection of texts like the modern New Testament is about 200 AD, a century after John was written. Of course the 4 main gospels weren't officially canon until 1545 as any fule kno.
( rjmrjm, Sun 22 Mar 2009, 20:21, Reply)