First I would like to thank all who contributed to the question, in particular The Photon, who led me to articles that gave a resolution of the problem.
Here is my take on the idea of forward and reflected waves on a feeder from an r.f. generator to a load.
I will use "c" for the speed of electromagnetic waves along the feeder.
I think the clue to reconciling the apparently different views on this is to consider what happens when a sine wave is initially applied by the transmitter to the feeder. The front travels down the line at c, with voltage and current in phase. What happens next depends on the termination of the line. If it is the characteristic (resistive) impedance Z0 of the line then there is no phase change, and the energy conveyed by the line is dissipated as heat, or radiated from the antenna (or a bit of both?) and this situation continues indefinitely.
If the line is not terminated in Z0 then there is a phase change due to this mismatch, and the effect of this travels back at c as the front of a reverse, reflected wave. Until this front reaches the transmitter, energy is still being sent into the feeder as a sine wave with voltage and current in phase.
When this returning front reaches the transmitter, the line impedance is no longer seen as Z0 because of this phase change. So there is a further reflection forward down the feeder, which is reflected back again at the load, and so on. This eventually results in a steady state, with a reduced forward flow of energy towards the load, and (in principle) an infinite number of counter-travelling waves. The only observable effect of this summation of forward and reflected waves is to change the input impedance of the line. The process is analysed and explained in:
Overall this is an excellent, fairly mathematical, article that dispels a lot of the myths and misconceptions common for this subject.
To quote from this article, for a normal, inevitably lossy, line "This derivation shows that the “standard” forward and reflected waves take a finite, though small, time to form, since energy has to travel up and down the line to create the wave assemblages that are summed. However, this process can be pretty well considered “instantaneous” at h.f., since typically a maximum of 10 or so line-lengths are travelled before contributions become vanishingly small - about half a microsecond on a typical 10 metre length of coax having a velocity factor of 66%."
In the resulting steady state, no further energy flows in the reverse direction, from load to transmitter. If we do think of there being a reflected wave then we can say that the energy it carries is subtracted from the forward energy that would have flowed, if the line was terminated in Z0, to give the net flow, which is always from transmitter to load. The transmitter never sends out more than the difference between the notional forward and reflected powers.
The steady state has a standing wave established on the feeder, governed by the situation at the load end. What the transmitter sees is determined by the length of the feeder, and depends on the phase relation at that point. At a voltage node the voltage is high and the current is low, and the transmitter sees the line as a high impedance. At a voltage anti-node it is the opposite, low voltage and high current, so the line is seen as a low impedance. We note that the standing wave is of average, usually r.m.s., values.
The standing wave can be resolved mathematically into forward and reflected components, also in the form of waves. In the forward component the voltage and current are in phase, and in the reverse component they are 180 degrees out of phase.
The current in the r.f. wave travelling forward can be analysed using phasors, into two components, that part (forward) in phase with the voltage, and that part (reflected) 180° out of phase. By sampling these currents they can be measured as voltages, and a calculation made of energy flow, i.e. forward and reflected powers, which can be displayed on moving coil meters with suitable calibrated scales. In order to use d.c. meters to measure the r.f. voltages, rectifying diodes are commonly found in SWR meters, nothing to do with separating the forward and reflected components. There area number of caveats to the power calculation, but for all practical purposes it is near enough.
To go further, for a less mathematical approach, I refer you to this article by Bruene, which nicely explains it all using graphical phasors: http://kambing.ui.ac.id/onnopurbo/orari-diklat/teknik/arrl/using-equipment/5904024.pdf
Note: My references worked on 23rd May 2013. I have found that some other references to these articles are no longer valid.