Alfonso says...

To distinguish the three meanings of 'time' I will re-express the set I
in the following not unnatural ways:
Set III
(a) The journey occurred in eternity.
(b) The instant of starting was 1 o'clock.
(c) The duration of the journey was 2 hours." Dingle

Note that only (c) has units of seconds. I think that part of the
problem is that we are all familiar with clocks# and think of them as
something which tells time in hours minutes and seconds. In scientific
terms we should perhaps not use the term clock but "duration meter" -
envisaging something with a digital reading which increments at some
interval 1/10^n seconds. The larger n then the better the resolution -
which is started by one detected event and stopped by another event.
Your statement:
"What time is, is what a clock locally at rest measures".
Becomes
What time is, is what a "duration meter" locally at rest measures.
In terms of my scenario the only interval which the duration meter can
measure is the interval between the ticks - the reciprocal of which is
the frequency of the ticks.

Right. Time in the sense of (b), which is a way of assigning
labels to events, requires two things: (1) a clock for measuring
durations, and (2) an initial setting.

The Lorentz transformations involve both.

The Doppler shift involve only duration, though.
There are two events, e_1 and e_2 occurring at the
sender. There are two corresponding events, e_3 and e_4
occurring at the receiver, where e_3 = the event in which
the receiver gets the light signal from e_1, and
e_4 = the event in which the receiver gets the light
signal from e_2. The Doppler shift is about the ratio

T_r/T_s

where T_r = the time between e_3 and e_4, as measured
by the receiver's clock, and T_s = the time between
e_1 and e_2, as measured by the sender's clock.

--
Daryl McCullough
Ithaca, NY