The information is carried by the electromagnetic field between the conductor and its return path.
The physical layer sees potential difference between points, and current as being the information bearers, but the actual propagation is as an EM field.
When you apply a voltage across the ends of the circuit a wave front propagates thru the field at the local speed of light (Typically about 2C/3 in most communications circuits) and it is this that carries the information and causes the (typically very slight) redistribution of the electrons in the circuit.
How you chose to code information into modulated current or voltage is down to how you write the physical layer specification, sometimes voltage being above or below a threshold maps to a 1 or 0 in the logic, sometimes currents, sometimes the presence of absence of an edge transition, sometimes OFDM of a whole block of bits, there are a lot of different ways to do it.
It is interesting to calculate the mean drift velocity of the electrons in a copper conductor at say 10mA in a 0.2mm^2 conductor cross section, you will be shocked at how low it is.
'Fields and waves in communications electronics' is the book you really want for this stuff, a nice excuse to get your math on.
Regards, Dan.