I think that we have an example of modern usage versus traditional usage. I have never previously heard of the Clacton line being described as a branch line, nor for that matter the Chingford line. As others have said, the traditional usage of the term "branch line" is for a relatively short, usually single track, line with a self containbed service requiring a change of trains at the main line junction. As for lines that don't join anything at the outer end, that makes anything with a terminus a branch line (Penzance, Holyhead etc.). The Clacton line is an integral part of the GEML service and always has been, so has never been considered as a branch line.