With transmission lines, especially those of substantial length, the voltage and current are
local, and don't include uncontrolled long wires outside the measurement.
Ideally, one defines a plane, perpendicular to the long direction of the coaxial cable (in this example) and the ideal voltage probe/oscilloscope measures the voltage from center to outer conductors in that plane, and the ideal current probe measures the current through the inner conductor passing through that plane.
Practical oscilloscope and current probes try to emulate that ideal situation.
When the diameter of the coax (again, in this example) becomes too large at the time scale of the measurement, we are crossing into waveguide territory, where we concentrate on the fields in the interior of the guide. Note that waveguides can be built with dielectric walls instead of conductive walls, with well-defined (in conventional electrodynamic theory) results similar to conductive guides, with the possibility of having a large DC (or lower-frequency than the waves) voltage from one end to the other. Similarly, if the interior of the waveguide is evacuated, one can run current in the form of electron or ion beams down the length of the guide, which will interact with the high-frequency fields. As mentioned in someone else's textbook citation, at the frequencies where waveguides are common, the skin depth in the metal walls is very small, and the energy is transported mainly in the
traveling wave down the guide. With a bad termination, or a resonant cavity, energy can be stored in the
standing wave.
There is a large literature and industrial history of both waveguides in general and particle accelerators in particular. Again, they both work.
When Heaviside made his huge breakthrough (realizing that adding lumped inductors in series with the telegraph lines improved their bandwidth), he may have used the telegraph system grounding as the return (bandwidth is not so high with manual Morse code). When AT&T successfully adopted his method to trunk telephone lines, they used twisted-pair transmission lines (balanced with respect to ground).
By the way, with respect to the outer insulation (jacket) on coaxial cables. I have used precision coaxial cables, some of which had armored jackets to prevent damage, and coaxial cables with various dielectrics (usually PE or PTFE), and various jackets (usually PVC), but also "semi-rigid" coaxial cables, where the outer conductor is essentially a copper tube, with no outer insulation. Careful use of these semi-rigid cables requires proper tooling for making bends, so as not to destroy the inner geometry. See
https://www.pasternack.com/pages/Featured_Products/hand-formable-semi-rigid-cable-assemblies-up-to-18-ghz-new-from-pasternack.html?utm_campaign=usa_cable_assemblies&keyword=semi-rigid%20coaxial%20cable%20assemblies&gclid=eaiaiqobchmi2dv1-dzv9wivshrnch1uia-oeaayasaaegkoypd_bwe for such assemblies available with or without outer insulating jacket. Again, they work the same either way.