My understanding is that the desired mode of operation for a two wire transmission line is differential, where at any point along the line the addition of the voltage or current is zero, such that the electric fields cancel and net radiation is zero.
Due to skin effect, in coax, the two wires are the inside of the braid and the outside of the center conductor. Technically speaking in this case the outside of the braid isn't part of the two wire transmission line.
For transmission lines, if you add together together the voltage or current at any point along the line and then divide by two, the result is the common mode voltage or current, which is that portion which does not have an equal and opposite. Common mode current inside coax can only be caused i believe by irregularities in the physical properties of the coax, or if the transmitter output isn't exactly differential. Common mode current is not caused by antennas which aren't balanced.
For all antennas fed with coax, at the junction between coax and antenna, current flowing away from the transmitter towards the antenna, will, due to skin effect, split up between the side of the antenna connected to the braid and the outside of the coax. The ratio of current flow between antenna and outside of coax is directly proportional to the ratio of the impedance wrt ground of the antenna element connected to the braid and the outside of the braid.
If it so happens that the impedance of the antenna element is low compared to that of the braid, then most of the current will flow into the antenna.
Common mode current inside of coax and current flowing on the outside of coax are two completely different conditions. For differential systems, common mode refers to an error condition concerning two wires, the two wires in this case being inside the coax. Current flowing on the outside of coax is a direct result of the voltage divider which exists at the junction of the braid of the coax and the connected antenna element, and is not common mode current at all. The outside of the braid is effectively part of a single ended single wire circuit with ground as the common point.
So why do ham radio operators refer to current on the outside of coax as common mode current, when quite clearly it isn't ?
To me it seems they are getting the outside of the braid mixed up as part of the two wire transmission line inside the coax, and this just serves to confuse everyone.