Cecil wrote,
"The forward current is equal at both ends of the coil. The reflected
current is equal at both ends of the coil."

If that's really true, then the net current is precisely equal at both
ends of the coil. I thought you had been claiming that the current is
different at each end. Which way is it going to be? If they are
different phases, then they are NOT equal. If they are different
phases, where does the phase shift COME FROM? If I allow a wave in one
direction ONLY and the currents at the two ends are DIFFERENT in phase,
WHAT HAPPENS inside the coil to make them different? Where does the
extra charge come from, or go to?

It's all very simple. Yawn.

Hint: Replace the coil with a piece of coaxial transmission line,
formed into a loop so the input and output ends are adjacent. Short
the outer conductors together and notice that nothing changes in terms
of the voltages across each end of the line and currents in the center
conductors at each end. Note the difference in current at the two ends
of the line, and note the current in the single outer conductor
terminal of this three-terminal system. Notice that the sum of all
three currents at every instant in time is essentially zero (current
direction taken as positive going into each terminal). Got it yet? Do
you understand WHAT it is, besides the inductance, that allows a coil
to look like a transmission line? Do you understand that the mode is
not quite TEM, so some of the usual TEM transmission line behaviour is
not going to hold?

Cheers,
Tom